Solving the Advection Diffusion Reaction Equations by Using the Enhanced Higher-Order Unconditionally Positive Finite Difference Method

Abstract

In this paper, the enhanced higher-order unconditionally positive finite difference method is developed to solve the linear, non-linear and system advection diffusion reaction equations. Investigation into the effectiveness and efficiency of the proposed method is carried out by calculating the convergence rate, error and computational time. A comparison of the solutions obtained by the enhanced higher-order unconditionally positive finite difference and exact solution is conducted for validation purposes. The numerical results show that the developed method reduced the time taken to solve the linear and non-linear advection diffusion reaction equations as compared to the results obtained by the higher-order unconditionally positive finite difference method.

1. Introduction

Numerical methods typically require large mesh sizes to give better approximations to solutions of differential equations. Consequently, their memory requirements are huge and, therefore, computational efficiency is affected. In this work, we aim to improve the computational efficiency of the higher-order unconditionally positive finite difference method (HUPFD). The HUPFD is based on using higher-order finite difference schemes to solve differential equations, with its main attribute being the preservation of the positivity of solutions. The positivity of solutions is essential in several physical settings—for example, quantities like population sizes, chemical species concentration and neutron numbers require positive solutions in order to be meaningful.

One of the techniques researchers employ to reduce the computational span of computational methods is the proper orthogonal decomposition method (POD) [1]. This has previously been used to reduce the dimensions of numerical methods such as non-standard finite difference and Crank–Nicolson to solve problems such as the Sobolev, hyperbolic, Burgers and Navier–Stokes equations [2,3,4]. In this paper, we couple the POD and HUPFD to obtain a new enhanced hybrid method that we name the enhanced higher-order unconditionally positive finite difference method (EHUPFD). By utilizing the POD, the EHUPFD extracts a set of basis functions from the HUPFD solution called the snapshot matrix and then uses a small subset of leading basis functions to construct state variable approximations.

We test the applicability of the EHUPFD on the advection diffusion reaction equation below

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u}{\partial t}}+U_a{\displaystyle \frac{\partial u}{\partial x}}-D{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=f(t,x,u),a\le x\le bt>0,x\in \partial \Omega \hfill \\ \hfill & u(0,x)=u_0\left(x\right)\ge 0,\hfill \\ \hfill & u(a,t)=u_a\left(t\right)u(b,t)=u_b\left(t\right),\hfill \end{array}$$

(1)

where D is the diffusion coefficient, $U_a$ is the advection coefficient, $u_x(t,x)$ is the advection term, $u_{xx}(t,x)$ is the diffusion term, $u_t(t,x)$ represent the rate of change of concentration of substances and $f(t,x,u)$ is the source or reaction term [5]. The ADR equation governs the process of diffusion and advection simultaneously [6]. The ADR equation is used to model the exponential traveling waves semiconductor modeling, biological system modeling, pollutant absorption in soil and neutron diffusion [6,7,8,9,10]. The UPFD method has previously been used to solve the ADR equation, amongst other known numerical methods reported in the literature, such as the standard finite difference, Crank–Nicolson finite difference, and non-standard finite difference methods and acts. The UPFD method has been proven to be a stable numerical method, accurate and consistent when solving the ADR equation in rectangular domains. The method is also known to preserve the positivity of solution regardless of step size [11]. For example, Appadu, A and Rao used UPFD to approximate the solution to the linear ADR equation, which models exponential traveling waves. The authors studied various scenarios in which advection or diffusion is dominating, as well as when both are equal. In this study, they discovered that the UPFD scheme is successful in solving the ADR equation when advection or diffusion factors are dominating [12]. However, as always, there is a need to constantly refine the numerical methods in order to improve the method’s accuracy, computational time and convergence rate.

The HUPFD method was developed to solve the advection diffusion reaction equations to improve the solution’s accuracy [13]. The higher-order numerical methods have been used to solve mathematical problems such as [13,14,15,16,17,18,19,20]. The standard finite difference methods, such as the backward, forward and central difference schemes, struggle to approach the exact solution of most partial differential models with large step sizes. As a result, relatively small step sizes are chosen to improve the accuracy of the solution to the problem at the cost of increased computational time due to the increased number of grid points. With a smaller grid size, HUPFD calculates a more accurate solution to the ADR equation’s linear, non-linear and system versions compared to the UPFD results.

The highly accurate finite difference method has gained much attention in recent years [21,22,23,24,25]. Gemechis FD and Tesfaye AB studied the higher-order numerical method to solve the singularly perturbed delay reaction–diffusion equation [26]. In order to achieve the six higher-order schemes, they used the Richardson extrapolation technique [26]. They found that the developed method is stable and consistent, and produces a more accurate solution than some of the methods reported in the literature. Ranjana KM and Venn G investigated the fourth-order finite difference method based on spline in tension approximation for the solution of second-order quasi-linear hyperbolic equations in one space dimension [27]. They proposed a three-level implicit nine-point compact finite difference formulation of order two in the time and four in the space directions based on split tension approximation in the t-direction for the numerical solution of space dimensional second-order quasi-linear hyperbolic equations with a first-order space derivative term. The proposed numerical method for the wave equation in polar coordinates is conditionally stable [27]. In contrast, the method for the damped wave and telegraphic equations is unconditionally stable. Sari M et al. [28] proposed a tenth-order finite difference scheme for solving a one-dimensional diffusion equation. Gurarslan et al. [29] presented a sixth-order compact difference scheme and a fourth-order Runge–Kutta scheme in time to produce numerical solutions of the one-dimensional advection–diffusion equation, which was found to be accurate in solving the concurrent transport equation for the one-dimensional advection diffusion reaction equation.

The rest of the sections in the paper are organized as follows. Section 2 and Section 3 introduce the UPFD and HUPFD, respectively. Subsequently, Section 4 and Section 5 apply the schemes in Section 2 and Section 3 to solve the ADR equations. Section 6 introduces the POD technique used to enhance the HUPFD, which results in the development of EHUPFD. The results obtained by the EHUPFD are illustrated in Section 6 and a summary of the results is provided in Section 7.

2. The Unconditionally Positive Finite Difference Scheme

The UPFD schemes are presented below [30],

$$\frac{\partial u}{\partial x}\approx \{\begin{array}{}\mathrm{(2a)}& \frac{u_i^n-u_{n-1}^{n+1}}{\Delta x}\mathrm{(2b)}& \frac{u_i^{n+1}-u_{n-1}^n}{\Delta x}\end{array}$$

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial u}{\partial t}}& \approx {\displaystyle \frac{u_i^{n+1}-u_i^n}{\Delta t}},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}& \approx {\displaystyle \frac{u_{i+1}^n-2u_i^{n+1}+u_{i-1}^n}{{(\Delta x)}^2}},\hfill \end{array}$$

(4)

where grid points are calculated by using the formula $x_i=a+\Delta x(i-1)$ for $1\le i\le m$ and spatial step size $\Delta x={\displaystyle \frac{b-a}{m-1}}$. The time step is calculated by using the formula $\Delta t={\displaystyle \frac{T}{N-1}}$ with grid points given by $t_n=\Delta t(n-1)$, for $1\le n\le N$. Equation (2a) is used when the coefficient of the first space derivative is negative, whereas Equation (2b) is used when the first space derivative term is positive. Equation (4) is used when the second space derivative is negative or positive.

3. The Higher-Order Unconditionally Positive Finite Difference Schemes for Coupling with POD

In this section, we present the HUPFD that will be coupled with the POD to develop the EHUPFD. The EHUPFD is then tested on the ADR equations.

HUPFD

In this case, we consider the first-order formula (2a) and (2b) for the first space derivative, (3) for the first time derivative, and the fourth order for the second space derivative approximations to obtain the higher-order schemes as given below,

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}& \approx {\displaystyle \frac{-u_{i+2}^{n+1}+16u_{i+1}^n-30u_i^{n+1}+16u_{i-1}^n-u_{i-2}^{n+1}}{12\Delta x^2}},\hfill \end{array}$$

(5)

where the spatial step size $\Delta x={\displaystyle \frac{b-a}{m-1}}$ and the grid points are given as $x_i=a+\Delta x(i-1)$ and $2\le i\le m+1$. The time step size $\Delta t={\displaystyle \frac{T}{N-1}}$, with the corresponding grid points given as $t_n=\Delta t(n-1)$, $1\le n\le N$.

In the development of the higher-order unconditionally positive finite difference (HUPFD), the goal is to produce a higher-order method, based on the principles of the UPFD, that will still preserve the positivity of solutions. We test the effectiveness of the scheme on the ADR equations. In practice, selecting different orders in finite difference schemes for space derivatives involves a tradeoff between accuracy, stability, consistency and computational cost. Engineers and scientists choose different orders of the schemes based on the specific requirements of their simulations and the constraints of their computational resources, which is often about achieving accuracy. In this study, the HUPFD schemes are constructed by using different orders in finite difference schemes for the space derivative scheme with the idea of investigating a combination that gives better results.

4. Implementation of the HUPFD Schemes on the Linear ADR Equation

In this section, we set $f(t,x,u)=-u$, $U_a=1$ and $D=1$ in Equation (1) to obtain the following equation

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{\partial u}{\partial x}}={\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}-u,\end{array}$$

(6)

The boundary and initial conditions used in this problem are given below

$$\begin{array}{c}\left\{\begin{array}{ll}u(x,0)=e^{-x},& 0\le x\le 10\\ u(0,t)=e^t,\hfill & 0\le t\le 0.85\\ u_x(10,t)=-u(10,t),\hfill & 0\le t\le 0.85\end{array}\right.\end{array}$$

(7)

with the exact solution given by $u(x,t)=e^{(t-x)}$ [30].

4.1. Solving the Linear ADR Equation Using the HUPFD Scheme

Because the coefficient of ${\displaystyle \frac{\partial u}{\partial x}}$ in Equation (6) is positive, we find the UPFD scheme using Equations (2b), (3) and (5). As a result, the following equation gives the HUPFD discretization of Equation (6)

$$\begin{array}{c}\hfill {\displaystyle \frac{u_i^{n+1}-u_i^n}{\Delta t}}+{\displaystyle \frac{u_i^{n+1}-u_{i-1}^n}{\Delta x}}={\displaystyle \frac{-u_{i+2}^{n+1}+16u_{i+1}^n-30u_i^{n+1}+16u_{i-1}^n-u_{i-2}^{n+1}}{12\Delta x^2}}-u_i^{n+1},\end{array}$$

(8)

which simplifies to the following

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{12\Delta x^2}}{u}_{i-2}^{n+1}& +\left({\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{1}{\Delta x}}+{\displaystyle \frac{30}{12\Delta x^2}}+1\right)u_i^{n+1}\hfill \\ \hfill & +{\displaystyle \frac{1}{12\Delta x^2}}{u}_{i+2}^{n+1}={\displaystyle \frac{1}{\Delta t}}{u}_i^n+\left({\displaystyle \frac{1}{\Delta x}}+{\displaystyle \frac{16}{12D{x}^2}}\right)u_{i-1}^2+{\displaystyle \frac{16}{12\Delta x^2}}{u}_{i+1}^n,\hfill \end{array}$$

(9)

let $p_1={\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{1}{\Delta x}}+{\displaystyle \frac{30}{12\Delta x^2}}+1$,$p_2={\displaystyle \frac{1}{12\Delta x^2}}$,$p_3={\displaystyle \frac{1}{2\Delta t}}$, $p_4={\displaystyle \frac{1}{\Delta x}}+{\displaystyle \frac{16}{12\Delta x^2}}$, thus

$$\begin{array}{c}\hfill p_1{u}_i^{n+1}+p_2{u}_{i+2}^{n+1}+p_2{u}_{i-2}^{n+1}=p_3{u}_i^n+p_4{u}_{i-1}^n+16p_2{u}_{i+1}^n,\end{array}$$

(10)

Equation (10) is simplified to the following

$$\begin{array}{c}\hfill A{u}^{n+1}=B{u}^n+d^n,\end{array}$$

(11)

where

$$\begin{array}{c}\hfill A{u}^{n+1}=\left[\begin{array}{ccccccc}{p}_1& 0& p_2& 0& 0& \cdots & 0\\ 0& p_1& 0& p_2& 0& \cdots & ⋮\\ p_2& 0& p_1& 0& p_2& \ddots & ⋮\\ 0& p_2& 0& p_1& 0& \ddots & 0\\ 0& 0& p_2& 0& p_1& \ddots & p_2\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & 0\\ 0& \cdots & \cdots & \cdots & p_2& 0& p_1\end{array}\right]\left[\begin{array}{c}{u}_1^{n+1}\\ u_2^{n+1}\\ u_3^{n+1}\\ ⋮\\ ⋮\\ ⋮\\ u_{M-1}^{n+1}\\ u_M^{n+1}\end{array}\right],\end{array}$$

(12)

$$\begin{array}{c}\hfill B{u}^n=\left[\begin{array}{ccccccc}{p}_3& 16p_2& 0& 0& 0& \cdots & 0\\ p_4& p_3& 16p_2& 0& 0& \cdots & ⋮\\ 0& p_4& p_3& 16p_2& 0& \ddots & ⋮\\ 0& 0& p_4& p_3& 16p_2& \ddots & 0\\ 0& 0& 0& p_4& p_3& \ddots & 0\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & 16p_2\\ 0& \cdots & \cdots & \cdots & 0& p_4& p_3\end{array}\right]\left[\begin{array}{c}{u}_1^n\\ u_2^n\\ u_3^n\\ ⋮\\ ⋮\\ ⋮\\ u_{M-1}^n\\ u_M^n\end{array}\right],\end{array}$$

(13)

and

$$\begin{array}{c}\hfill d^n=p_3\left[\begin{array}{c}{u}_0^n\\ 0\\ 0\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ u_M^n\end{array}\right]-p_2\left[\begin{array}{c}{u}_0^{n+1}\\ 0\\ 0\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ u_{M+1}^{n+1}\end{array}\right],\end{array}$$

(14)

where u is the unknown vector of the order $M\times 1$ at any time level $t^{n+1}$, and A and B are the coefficient square matrix of order $M\times M$ with a triangular structure.

4.2. Solving the Non-Linear ADR Equation Using the HUPFD Scheme

In this section, the EHUPFD is used to solve the non-linear advection diffusion reaction equation given below, with $f(t,x,u)=ru(1-u)$ and $U_a=0$

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial u}{\partial t}}-D{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=ru(1-u),\end{array}$$

(15)

The boundary and initial conditions of (15) are given below

$$\begin{array}{c}\hfill u(x,0)=\left\{\begin{array}{c}0.7e^{-\sigma {(x-x_0)}^2},|x-x_0|\le 0.06\hfill \\ 0,\mathrm{otherwise}\hfill \end{array}\right.\end{array}$$

(16)

$$\begin{array}{cc}\hfill u(0,t)& =0,0\le t\le T\hfill \\ \hfill u(1,t)& =0,\hfill \end{array}$$

where $\sigma =80$ and $x_0=0.5$ [30].

The exact solution is given as follows

$$\begin{array}{c}\hfill u(t,x)={\displaystyle \frac{1}{{\left[1+\left({\displaystyle \frac{1-\sqrt{u(x,0)}}{\sqrt{u(x,0)}exp\left({\displaystyle \frac{1}{6D}}\sqrt{6Dr}x\right)}}\right)exp\left(-{\displaystyle \frac{5}{6}}rt+{\displaystyle \frac{1}{6D}}\sqrt{6Dr}x\right)\right]}^2}},\end{array}$$

(17)

where $D=0.0002$ and $r=0.05$.

The discretization of the non-linear ADR using Equations (2b), (3) and (15) is given below

$$\begin{array}{c}\hfill {\displaystyle \frac{u_i^{n+1}-u_i^{n-1}}{2\Delta t}}-D{\displaystyle \frac{-u_{i+2}^{n+1}+16u_{i+1}^n-30u_i^{n+1}+16u_{i-1}^n-u_{i-2}^{n+1}}{12\Delta x^2}}=r{u}_i^n-u_{max}{u}_i^{n+1},\end{array}$$

(18)

where $u_{max}$ is a frozen coefficient. By simplifying Equation (18), the following is obtained

$$\begin{array}{c}\left({\displaystyle \frac{1}{2\mathsf{\Delta }t}}+{\displaystyle \frac{30}{12{\left(\mathsf{\Delta }x\right)}^2}}+r{u}_{max}\right)u_i^{n+1}+{\displaystyle \frac{D}{12{\left(\mathsf{\Delta }x\right)}^2}}\left(u_{i+2}^{n+1}+u_{i-2}^{n+1}\right)={\displaystyle \frac{D}{12{\left(\mathsf{\Delta }x\right)}^2}}\left(16u_{i+1}^n+16u_{i-1}^n\right)+r{u}_i^n+{\displaystyle \frac{1}{2\mathsf{\Delta }t}}{u}_i^{n-1},\hfill \end{array}$$

Let $s_1={\displaystyle \frac{1}{2\Delta t}}+{\displaystyle \frac{30}{12\Delta x}}+r{u}_{max}$, $s_2={\displaystyle \frac{D}{12{(\Delta x)}^2}}$ and $s_3={\displaystyle \frac{1}{2\Delta t}}$; thus,

$$\begin{array}{c}\hfill s_2{u}_{i-2}^{n+1}+s_1{u}_i^{n+1}+s_2{u}_{i+2}^{n+1}=16s_2{u}_{i-1}^n+r{u}_i^n+16s_2{u}_{i+1}^n+s_3{u}_i^{n-1},\end{array}$$

(19)

Equation (19) is simplified to the following equation

$$\begin{array}{c}\hfill A{u}^{n+1}=B{u}^n+d^n\end{array}$$

(20)

where

$$\begin{array}{c}\hfill A{u}^{n+1}=\left[\begin{array}{ccccccc}{s}_1& 0& s_2& 0& 0& \cdots & 0\\ 0& s_1& 0& s_2& 0& \cdots & ⋮\\ s_2& 0& s_1& 0& s_2& \ddots & ⋮\\ 0& s_2& 0& s_1& 0& \ddots & 0\\ 0& 0& s_2& 0& s_1& \ddots & s_2\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & 0\\ 0& \cdots & \cdots & \cdots & \cdots & s_2& s_1\end{array}\right]\left[\begin{array}{c}{u}_1^{n+1}\\ u_2^{n+1}\\ u_3^{n+1}\\ ⋮\\ ⋮\\ ⋮\\ u_{M-1}^{n+1}\\ u_M^{n+1}\end{array}\right],\end{array}$$

(21)

$$\begin{array}{c}\hfill B{u}^n=\left[\begin{array}{ccccccc}r& 16s_2& 0& 0& 0& \cdots & 0\\ 16s_2& r& 16s_2& 0& 0& \cdots & ⋮\\ 0& 16s_2& r& 16s_2& 0& \ddots & ⋮\\ 0& 0& 16s_2& r& 16s_2& \ddots & 0\\ 0& 0& 0& 16s_2& r& \ddots & 0\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & 16s_2\\ 0& \cdots & \cdots & \cdots & 0& 16s_2& r\end{array}\right]\left[\begin{array}{c}{u}_1^n\\ u_2^n\\ u_3^n\\ ⋮\\ ⋮\\ ⋮\\ u_{M-1}^n\\ u_M^n\end{array}\right],\end{array}$$

(22)

and

$$\begin{array}{c}\hfill d^n=s_3\left[\begin{array}{c}{u}_1^n\\ u_2^n\\ u_3^n\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ u_{M-1}^n\\ u_M^n\end{array}\right].\end{array}$$

(23)

4.3. Implementation of the UPFD and the HUPFD Schemes on the Linear Schnakenberg Model

The linear Schnakenberg model is given by the equation below

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial u}{\partial t}}+c{\displaystyle \frac{\partial u}{\partial x}}& =(q_1-1)u+(q_2+s)v+{\mu }_1{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial t}}+c{\displaystyle \frac{\partial v}{\partial x}}& =-q_{1}u-q_{2}v+{\mu }_2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}},\hfill \end{array}$$

(25)

subject to the boundary and initial conditions given below

$$\begin{array}{c}\hfill \left\{\begin{array}{c}u(x,0)=e^{-x},0\le x\le 10\hfill \\ u(0,t)=e^t,0\le t\le 0.85\hfill \\ u_x(10,t)=-u(10,t),0\le t\le 0.85\hfill \end{array}\right.\end{array}$$

(26)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}v(x,0)=e^{-x},0\le x\le 10\hfill \\ v(0,t)=e^t,0\le t\le 0.85\hfill \\ v_x(10,t)=-v(10,t),0\le t\le 0.85\hfill \end{array}\right.\end{array}$$

(27)

with constants $s=5,c=0,q_1=q_2=0,{\mu }_1=1$ and ${\mu }_2=5$.

4.3.1. Solving the Linear Schnakenberg Model Using the UPFD Scheme

The UPFD scheme is applied to the linear Schnakenberg model and the discretization of the scheme is given below

$$\begin{array}{cc}\hfill {\displaystyle \frac{u_i^{n+1}-u_i^n}{\Delta t}}+c{\displaystyle \frac{u_i^{n+1}-u_i^n}{\Delta x}}& =(q_1-1)u_i^{n+1}+(q_2+s)v_i^n+{\mu }_1\left({\displaystyle \frac{u_{i+1}^n-2u_i^{n+1}+u_{i-1}^n}{{(\Delta x)}^2}}\right),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\displaystyle \frac{v_i^{n+1}-v_i^n}{\Delta t}}+c{\displaystyle \frac{v_i^{n+1}-v_i^n}{\Delta x}}& =-q_1{u}_i^{n+1}-q_2{v}_i^{n+1}+{\mu }_2\left({\displaystyle \frac{v_{i+1}^n-2v_i^{n+1}+v_{i-1}^n}{{(\Delta x)}^2}}\right),\hfill \end{array}$$

(29)

which simplifies to the following

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{c}{\Delta x}}+(q_1-1)+{\displaystyle \frac{2{\mu }_1}{{(\Delta x)}^2}}\right)u_i^{n+1}=(q_2+s)v_i^n& +{\displaystyle \frac{{\mu }_1}{\Delta x}}(u_{i+1}^n+u_{i-1}^n)+{\displaystyle \frac{c}{\Delta x}}{u}_i^n\hfill \\ \hfill & +{\displaystyle \frac{1}{\Delta t}}{u}_i^n,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{c}{\Delta x}}+q_2+{\displaystyle \frac{2{\mu }_2}{{(\Delta x)}^2}}\right)u_i^{n+1}=-q_1{u}_i^{n+1}& +{\displaystyle \frac{1}{\Delta t}}{v}_i^n+{\displaystyle \frac{c}{\Delta x}}{v}_i^n\hfill \\ \hfill & +{\displaystyle \frac{{\mu }_2}{{(\Delta x)}^2}}(v_{i+1}^n+v_{i-1}^n),\hfill \end{array}$$

(31)

Let $s_1=\left({\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{c}{\Delta x}}+(q_1-1)+{\displaystyle \frac{2{\mu }_1}{{(\Delta x)}^2}}\right)$ and $s_2=\left({\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{c}{\Delta x}}+q_2+{\displaystyle \frac{2{\mu }_2}{{(\Delta x)}^2}}\right)$; thus,

$$\begin{array}{cc}\hfill u_i^{n+1}& ={\displaystyle \frac{(q_2+s)}{s_1}}{v}_i^n+{\displaystyle \frac{{\mu }_1}{\Delta x{s}_1}}(u_{i+1}^n+u_{i-1}^n)+{\displaystyle \frac{c}{\Delta x{s}_1}}{u}_i^n+{\displaystyle \frac{1}{\Delta t{s}_1}}{u}_i^n,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill u_i^{n+1}& =-{\displaystyle \frac{q_1}{s_2}}{u}_i^{n+1}+{\displaystyle \frac{1}{\Delta t{s}_2}}{v}_i^n+{\displaystyle \frac{c}{\Delta x{s}_2}}{v}_i^n+{\displaystyle \frac{{\mu }_2}{{(\Delta x)}^2{s}_2}}(v_{i+1}^n+v_{i-1}^n),\hfill \end{array}$$

(33)

4.3.2. Prove the Stability of the System ADR Equation with the UPFD

By discretizing (54) and (55) by using the UPFD scheme, we obtain the following

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{1}{\Delta t}}+(1-q_1)+{\displaystyle \frac{2{\mu }_1}{{(\Delta x)}^2}}\right)u_i^{n+1}+{\displaystyle \frac{c}{\Delta x}}{u}_{i+1}^{n+1}=q_2{v}_i^n& +{\displaystyle \frac{{\mu }_1}{{(\Delta x)}^2}}\left(u_{i+1}^n+u_{i-1}^n\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\Delta t}}{u}_i^n+{\displaystyle \frac{c}{\Delta x}}{u}_i^n,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill \left({\displaystyle \frac{1}{\Delta t}}+q_2+{\displaystyle \frac{2{\mu }_2}{{(\Delta x)}^2}}\right)v_i^{n+1}+{\displaystyle \frac{c}{\Delta x}}{v}_{i+1}^{n+1}=-q_1{u}_i^{n+1}& +{\displaystyle \frac{1}{\Delta t}}{v}_i^n+{\displaystyle \frac{c}{\Delta x}}{v}_i^n\hfill \\ \hfill & +{\displaystyle \frac{{\mu }_2}{{(\Delta x)}^2}}\left(v_{i+1}^n+v_{i-1}^n\right),\hfill \end{array}$$

(35)

By using the parameters $s=5,c=0,q_1=q_2=0,{\mu }_1=1$ and ${\mu }_2=5$, Equations (34) and (35) are simplified to

$$\begin{array}{cc}\hfill & \left({\displaystyle \frac{1}{\Delta t}}+1+{\displaystyle \frac{2}{{(\Delta x)}^2}}\right)u_i^{n+1}={\displaystyle \frac{1}{{(\Delta x)}^2}}\left(u_{i+1}^n+u_{i-1}^n\right)+{\displaystyle \frac{1}{\Delta t}}{u}_i^n\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & \left({\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{10}{{(\Delta x)}^2}}\right)v_i^{n+1}={\displaystyle \frac{1}{\Delta t}}{v}_i^n+{\displaystyle \frac{5}{{(\Delta x)}^2}}\left(v_{i+1}^n+v_{i-1}^n\right)\hfill \end{array}$$

(37)

The von Neumann stability analysis, $u_i^n={\xi }^n{e}^{j\Delta ti\Delta x}$, is used to check the stability region in Equations (36) and (37) to obtain the following

$$\begin{array}{c}\hfill \left({\displaystyle \frac{1}{\Delta t}}+1+{\displaystyle \frac{2}{{(\Delta x)}^2}}\right){\xi }^{n+1}{e}^{j\Delta ti\Delta x}={\displaystyle \frac{1}{{(\Delta x)}^2}}\left({\xi }^n{e}^{j\Delta t(i+1)\Delta x}+{\xi }^n{e}^{j\Delta t(i-1)\Delta x}\right)+{\displaystyle \frac{1}{\Delta t}}{\xi }^n{e}^{j\Delta ti\Delta x},\end{array}$$

(38)

$$\begin{array}{c}\hfill \left({\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{10}{{(\Delta x)}^2}}\right){\xi }^{n+1}{e}^{j\Delta ti\Delta x}={\displaystyle \frac{1}{\Delta t}}{\xi }^n{e}^{j\Delta ti\Delta x}+{\displaystyle \frac{5}{{(\Delta x)}^2}}\left({\xi }^n{e}^{j\Delta t(i+1)\Delta x}+{\xi }^n{e}^{j\Delta t(i-1)\Delta x}\right),\end{array}$$

(39)

Equations (38) and (39) are simplified to obtain the following

$$\begin{array}{c}\hfill \xi ={\displaystyle \frac{{\displaystyle \frac{2{\mu }_1}{{(\Delta x)}^2}}cos\left(\omega \right)+{\displaystyle \frac{1}{\Delta t}}}{{\displaystyle \frac{1}{\Delta t}}+1+{\displaystyle \frac{2}{{(\Delta x)}^2}}}},\end{array}$$

(40)

and

$$\begin{array}{c}\hfill \xi ={\displaystyle \frac{{\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{10cos\left(\omega \right)}{{(\Delta x)}^2}}}{{\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{10}{{(\Delta x)}^2}}}},\end{array}$$

(41)

where $\omega =\Delta t\Delta x$. By using the inequality $\left|\xi \right|\le 1$, when $\omega =\pi$, we obtain the following, respectively

$$\begin{array}{c}\hfill \Delta t>-2.\end{array}$$

(42)

4.3.3. Proof of Consistency with the System ADR Equation by Using UPFD Scheme

By discretizing (24) and (25) by using the UPFD scheme, we obtain the following

$$\begin{array}{cc}\hfill {\displaystyle \frac{u_i^{n+1}-u_i^n}{\Delta t}}+c{\displaystyle \frac{u_{i+1}^{n+1}-u_i^2}{\Delta x}}& =(q_1-1)u_i^{n+1}+(q_2+s)v_i^n+{\displaystyle \frac{{\mu }_1}{{(\Delta x)}^2}}\left(u_{i+1}^n-2u_i^{n+1}+u_{i-1}^n\right),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill {\displaystyle \frac{v_i^{n+1}-v_i^n}{\Delta t}}+c{\displaystyle \frac{v_{i+1}^{n+1}-v_i^2}{\Delta x}}& =-q_1{u}_i^{n+1}-q_2{v}_i^n+{\displaystyle \frac{{\mu }_2}{{(\Delta x)}^2}}\left(v_{i+1}^n-2v_i^{n+1}+v_{i-1}^n\right).\hfill \end{array}$$

(44)

From (43) and (44), we obtain the following Taylor’s expansions

$$\begin{array}{cc}\hfill u_i^{n+1}& \approx u_i^n+\Delta t{\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+{\displaystyle \frac{{(\Delta t)}^3}{6}}{\displaystyle \frac{{\partial }^{3}u}{\partial t^3}}+\cdots \hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill u_i^{n-1}& \approx u_i^n-\Delta t{\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-{\displaystyle \frac{{(\Delta t)}^3}{6}}{\displaystyle \frac{{\partial }^{3}u}{\partial t^3}}+\cdots \hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill u_{i-1}^n& \approx u_i^n-\Delta x{\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{{(\Delta x)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}-{\displaystyle \frac{{(\Delta x)}^3}{6}}{\displaystyle \frac{{\partial }^{3}u}{\partial x^3}}+\cdots \hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill u_{i+1}^n& \approx u_i^n+\Delta x{\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{{(\Delta x)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{(\Delta x)}^3}{6}}{\displaystyle \frac{{\partial }^{3}u}{\partial x^3}}+\cdots \hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill u_{i-2}^{n+1}& \approx u_i^n+\Delta t{\displaystyle \frac{\partial u}{\partial t}}-2\Delta x{\displaystyle \frac{\partial u}{\partial x}}+2{(\Delta x)}^2{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}-2(\Delta t\Delta x){\displaystyle \frac{{\partial }^{2}u}{\partial t\partial x}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+\cdots \hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill u_{i+2}^{n+1}& \approx u_i^n+\Delta t{\displaystyle \frac{\partial u}{\partial t}}+2\Delta x{\displaystyle \frac{\partial u}{\partial x}}+2{(\Delta x)}^2{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+2(\Delta t\Delta x){\displaystyle \frac{{\partial }^{2}u}{\partial t\partial x}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+\cdots \hfill \end{array}$$

(50)

By applying Taylor’s expansions to Equations (43) and (44), we obtain the following simplified equations

$$\begin{array}{cc}\hfill (1-q_1)u_i^n& +\left(1+(1-q_1)\Delta t{\displaystyle \frac{2{\mu }_1}{\Delta x^2}}\Delta t+{\displaystyle \frac{c\Delta t}{\Delta x}}\right){\displaystyle \frac{\partial u}{\partial t}}+c{\displaystyle \frac{\partial u}{\partial x}}\hfill \\ \hfill & +\left({\displaystyle \frac{\Delta t}{2}}+(1-q_1){\displaystyle \frac{{(\Delta t)}^2}{2}}+{\displaystyle \frac{2{\mu }_1}{{(\Delta x)}^2}}+{\displaystyle \frac{2{\mu }_1{(\Delta t)}^3}{6{(\Delta x)}^2}}+{\displaystyle \frac{c(\Delta t)}{\Delta x}}\right){\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\hfill \\ \hfill & +2c\Delta t{\displaystyle \frac{{\partial }^{2}u}{\partial t\partial x}}-q_2{v}_i^n-{\mu }_1{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=0,\hfill \end{array}$$

(51)

and

$$\begin{array}{c}\left({\displaystyle \frac{1}{\Delta t}}+q_2+{\displaystyle \frac{2{\mu }_2}{{(\Delta x)}^2}}+{\displaystyle \frac{c}{\Delta x}}\right)v_i^n+\left(1+q_2\Delta t+{\displaystyle \frac{2{\mu }_2\Delta t}{{(\Delta x)}^2}}+{\displaystyle \frac{c\Delta t}{\Delta x}}\right){\displaystyle \frac{\partial v}{\partial t}}+c{\displaystyle \frac{\partial v}{\partial x}}\hfill \\ +\left({\displaystyle \frac{\Delta t}{2}}+{\displaystyle \frac{q_2{(\Delta t)}^2}{2}}+{\displaystyle \frac{{\mu }_2{(\Delta t)}^2}{{(\Delta x)}^2}}+{\displaystyle \frac{c{(\Delta t)}^2}{2(\Delta x)}}\right){\displaystyle \frac{{\partial }^{2}v}{\partial t^2}}+2\Delta t{\displaystyle \frac{{\partial }^{2}v}{\partial t\partial x}}+{\displaystyle \frac{c(\Delta x)}{2}}{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}\\ =-1_1{u}_i^n+\left({\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{2{\mu }_2}{{(\Delta x)}^2}}\right)v_i^n-q_1\Delta t{\displaystyle \frac{\partial u}{\partial t}}-q_1{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\\ \hfill -{\displaystyle \frac{q_1{(\Delta t)}^3}{6}}{\displaystyle \frac{{\partial }^{3}u}{\partial t^3}}+{\mu }_2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}-{\displaystyle \frac{c}{\Delta x}}{v}_i^n,\end{array}$$

(52)

Let $\Delta t={(\Delta x)}^3$, which leads to Equations (51) and (52) being written as follows

$$\begin{array}{cc}\hfill (1-q_1)u_i^n& +\left(1+(1-q_1){(\Delta x)}^3+2{\mu }_1\Delta x+c{(\Delta x)}^2\right){\displaystyle \frac{\partial u}{\partial x}}\hfill \\ \hfill & +c{\displaystyle \frac{\partial u}{\partial x}}+\left({\displaystyle \frac{{(\Delta x)}^3}{2}}+(1-q_1){\displaystyle \frac{{(\Delta x)}^6}{2}}+{\displaystyle \frac{{\mu }_1{(\Delta x)}^4}{3}}+c{(\Delta x)}^2\right){\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\hfill \\ \hfill & +2c{(\Delta x)}^3{\displaystyle \frac{{\partial }^{2}u}{\partial t\partial x}}-q_2{v}_i^n-{\mu }_1{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}=0,\hfill \end{array}$$

and

$$\begin{array}{c}\hfill q_2{v}_i^n+q_1{u}_i^n+\left(1+q_2{(\Delta x)}^3+2{\mu }_2\Delta x+c{(\Delta x)}^2+q_1{(\Delta x)}^3\right){\displaystyle \frac{\partial v}{\partial t}}+c{\displaystyle \frac{\partial v}{\partial x}}\\ \hfill +\left({\displaystyle \frac{{(\Delta x)}^3}{2}}+q_2{\displaystyle \frac{{(\Delta x)}^6}{2}}+{\mu }_2{(\Delta x)}^4+d\frac{{(\Delta x)}^5}{2}+q_1{\displaystyle \frac{{(\Delta x)}^6}{2}}\right){\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\\ \hfill +2\Delta t{\displaystyle \frac{{\partial }^{2}v}{\partial t\partial x}}+\left({\displaystyle \frac{c(\Delta x)}{2}}-{\mu }_2\right)=0,\end{array}$$

(53)

When $\Delta x⟶0$, we obtain the original system ADR equations

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial u}{\partial t}}+c{\displaystyle \frac{\partial u}{\partial x}}& =(q_1-1)u+(q_2+s)v+{\mu }_1{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial t}}+c{\displaystyle \frac{\partial v}{\partial x}}& =-q_{1}u-q_{2}v+{\mu }_2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}},\hfill \end{array}$$

(55)

4.3.4. Solving the Schnakenberg Model Using the HUPFD Scheme

The HUPFD scheme is applied to the linear Schnakenberg model and the discretization of the scheme is given below

$$\begin{array}{cc}\hfill {\displaystyle \frac{u_i^{n+1}-u_i^n}{\Delta t}}& +c{\displaystyle \frac{u_i^{n+1}-u_{i-1}^n}{\Delta x}}=(q_1-1)u_i^{n+1}+(q_2+s)v_i^n,\hfill \end{array}$$

$$\begin{array}{cc}\hfill & +{\mu }_1\left({\displaystyle \frac{-u_{i+2}^{n+1}+16u_{i+1}^n-30u_i^{n+1}+16u_{i-1}^n-u_{i-2}^{n+1}}{12{(\Delta x)}^2}}\right),\hfill \\ \hfill {\displaystyle \frac{v_i^{n+1}-v_i^n}{\Delta t}}& +c{\displaystyle \frac{v_i^{n+1}-v_{i-1}^n}{\Delta x}}=-q_1{u}_i^{n+1}-q_2{v}_i^{n+1},\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & +{\mu }_2\left({\displaystyle \frac{-v_{i+2}^{n+1}+16v_{i+1}^n-30v_i^{n+1}+16v_{i-1}^n-v_{i-2}^{n+1}}{12{(\Delta x)}^2}}\right),\hfill \end{array}$$

(57)

which are simplified to the following

$$\begin{array}{c}\hfill d_1{u}_i^{n+1}+d_2{u}_{i+2}^{n+1}+d_2{u}_{i-2}^{n+1}=d_4{u}_i^n+(q_2+s)v_i^n+16d_2{u}_{i+1}^n+d_3{u}_{i-1}^n,\end{array}$$

(58)

$$\begin{array}{c}\hfill p_1{v}_i^{n+1}+p_2{v}_{i+2}^{n+1}+p_2{v}_{i-2}^{n+1}+q_1{u}_i^{n+1}=d_4{v}_i^{n+1}+p_3{v}_{i-1}^n+16p_2{v}_{i+1}^n,\end{array}$$

(59)

where $a_1={\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{c}{\Delta x}}+{\displaystyle \frac{30{\mu }_1}{12{(\Delta )}^2}}$, $a_2={\displaystyle \frac{{\mu }_1}{12{(\Delta x)}^2}}$, $a_3={\displaystyle \frac{c}{\Delta x}}+{\displaystyle \frac{16{\mu }_1}{12{(\Delta x)}^2}}$, $a_4={\displaystyle \frac{1}{\Delta t}}$, $p_1={\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{c}{\Delta x}}+q_2+{\displaystyle \frac{30{\mu }_2}{12{(\Delta x)}^2}}$, $p_2={\displaystyle \frac{{\mu }_2}{12{(\Delta x)}^2}}$ and $p_3={\displaystyle \frac{c}{\Delta x}}+{\displaystyle \frac{16{\mu }_2}{12{(\Delta x)}^2}}$

Equations (58) and (59) are simplified to the following

$$\begin{array}{c}\hfill A_1{u}^{n+1}=B_1{u}^n+d_1^n\mathrm{and}{A}_2{u}^{n+1}=B_2{u}^n+d_2^n,\mathrm{respectively}\end{array}$$

(60)

where

$$\begin{array}{c}\hfill A_1{u}^{n+1}=\left[\begin{array}{ccccccc}{a}_1& 0& a_2& 0& 0& \cdots & 0\\ 0& a_1& 0& a_2& 0& \cdots & ⋮\\ a_2& 0& a_1& 0& a_2& \ddots & ⋮\\ 0& a_2& 0& a_1& 0& \ddots & 0\\ 0& 0& a_2& 0& a_1& \ddots & a_2\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & 0\\ 0& \cdots & \cdots & \cdots & a_2& 0& a_1\end{array}\right]\left[\begin{array}{c}{u}_1^{n+1}\\ u_2^{n+1}\\ u_3^{n+1}\\ ⋮\\ ⋮\\ ⋮\\ u_{M-1}^{n+1}\\ u_M^{n+1}\end{array}\right],\end{array}$$

(61)

$$\begin{array}{c}\hfill B_1{u}^n=\left[\begin{array}{ccccccc}{a}_4& 16a_2& 0& 0& 0& \cdots & 0\\ a_3& a_4& 16a_2& 0& 0& \cdots & ⋮\\ 0& a_3& a_4& 16a_2& 0& \ddots & ⋮\\ 0& 0& a_3& a_4& 16a_2& \ddots & 0\\ 0& 0& 0& d_3& a_4& \ddots & 0\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & 16a_2\\ 0& \cdots & \cdots & \cdots & 0& a_3& a_4\end{array}\right]\left[\begin{array}{c}{u}_1^n\\ u_2^n\\ u_3^n\\ ⋮\\ ⋮\\ ⋮\\ u_{M-1}^n\\ u_M^n\end{array}\right],\end{array}$$

(62)

and

$$\begin{array}{c}\hfill d_1^n=a_2\left[\begin{array}{c}{u}_0^n\\ 0\\ 0\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ u_M^n\end{array}\right]-a_3\left[\begin{array}{c}{u}_0^{n+1}\\ 0\\ 0\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ u_{M+1}^{n+1}\end{array}\right],\end{array}$$

(63)

and

$$\begin{array}{c}\hfill A_2{v}^{n+1}=\left[\begin{array}{ccccccc}{p}_1& 0& p_2& 0& 0& \cdots & 0\\ 0& p_1& 0& p_2& 0& \cdots & ⋮\\ p_2& 0& p_1& 0& p_2& \ddots & ⋮\\ 0& p_2& 0& p_1& 0& \ddots & 0\\ 0& 0& p_2& 0& p_1& \ddots & p_2\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & 0\\ 0& \cdots & \cdots & \cdots & p_2& 0& p_1\end{array}\right]\left[\begin{array}{c}{v}_1^{n+1}\\ v_2^{n+1}\\ v_3^{n+1}\\ ⋮\\ ⋮\\ ⋮\\ v_{M-1}^{n+1}\\ v_M^{n+1}\end{array}\right],\end{array}$$

(64)

$$\begin{array}{c}\hfill B_2{v}^n=\left[\begin{array}{ccccccc}{d}_4& 16p_2& 0& 0& 0& \cdots & 0\\ p_3& d_4& 16p_2& 0& 0& \cdots & ⋮\\ 0& p_3& d_4& 16p_2& 0& \ddots & ⋮\\ 0& 0& p_3& d_4& 16p_2& \ddots & 0\\ 0& 0& 0& p_3& d_4& \ddots & 0\\ ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & 16p_2\\ 0& \cdots & \cdots & \cdots & 0& p_3& d_4\end{array}\right]\left[\begin{array}{c}{v}_1^n\\ v_2^n\\ v_3^n\\ ⋮\\ ⋮\\ ⋮\\ v_{M-1}^n\\ v_M^n\end{array}\right],\end{array}$$

(65)

and

$$\begin{array}{c}\hfill d_2^n=a_2\left[\begin{array}{c}{v}_0^n\\ 0\\ 0\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ v_M^n\end{array}\right]-a_3\left[\begin{array}{c}{v}_0^{n+1}\\ 0\\ 0\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ ·\\ v_{M+1}^{n+1}\end{array}\right],\end{array}$$

(66)

respectively, where u is the unknown vector of the order $M\times 1$ at any time level $t^{n+1}$, and A and B are the coefficient square matrix of order $M\times M$ with a triangular structure.

4.3.5. Prove Stability When Solving System ADR Equation by the HUPFD

By discretizing (56) and (57) by using the UPFD scheme, we obtain the following

$$\begin{array}{cc}\hfill {\displaystyle \frac{u_i^{n+1}-u_i^n}{\Delta t}}+c{\displaystyle \frac{u_{i+1}^{n+1}-u_i^n}{\Delta x}}& =(q_1-1)u_i^{n+1}+(q_2+s)v_i^n\hfill \\ \hfill & +{\mu }_1\left({\displaystyle \frac{-u_{i+2}^{n+1}+16u_{i+1}^n-30u_i^{n+1}+16u_{i-1}^n-u_{i-2}^{n+1}}{12{(\Delta x)}^2}}\right),\hfill \end{array}$$

(67)

$$\begin{array}{cc}\hfill {\displaystyle \frac{v_i^{n+1}-v_i^n}{\Delta t}}+c{\displaystyle \frac{v_{i+1}^{n+1}-v_i^n}{\Delta x}}& =-q_1{u}_i^{n+1}-q_2{v}_i^{n+1}\hfill \\ \hfill & +{\mu }_1\left({\displaystyle \frac{-v_{i+2}^{n+1}+16v_{i+1}^n-30v_i^{n+1}+16v_{i-1}^n-v_{i-2}^{n+1}}{12{(\Delta x)}^2}}\right),\hfill \end{array}$$

(68)

By using parameters $s=5,c=0,q_1=q_2=0,{\mu }_1=1$ and ${\mu }_2=5$, Equations (67) and (70) are simplified to

$$\begin{array}{cc}\hfill & \left({\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{30}{12{(\Delta x)}^2}}\right)v_i^{n+1}+{\displaystyle \frac{5}{12{(\Delta x)}^2}}\left(v_{i+2}^{n+1}+v_{i-1}^{n+1}\right)={\displaystyle \frac{1}{\Delta t}}{v}_i^n+{\displaystyle \frac{80}{12{(\Delta x)}^2}}\left(v_{i+1}^n+v_{i-1}^n\right),\hfill \end{array}$$

(69)

$$\begin{array}{c}{\displaystyle \frac{v_i^{n+1}-v_i^n}{\Delta t}}+c{\displaystyle \frac{v_{i+1}^{n+1}-v_i^n}{\Delta x}}=-q_1{u}_i^{n+1}-q_2{v}_i^{n+1}\hfill \\ \hfill +{\mu }_1\left({\displaystyle \frac{-v_{i+2}^{n+1}+16v_{i+1}^n-30v_i^{n+1}+16v_{i-1}^n-v_{i-2}^{n+1}}{12{(\Delta x)}^2}}\right),\end{array}$$

(70)

The von Neumann stability analysis, $u_i^n={\xi }^n{e}^{j\Delta ti\Delta x}$, is used to check the stability region in Equations (67) and (70) to obtain the following

$$\begin{array}{c}\hfill \left({\displaystyle \frac{1}{\Delta t}}+1-q_1+{\displaystyle \frac{30{\mu }_1}{12{(\Delta x)}^2}}\right){\xi }^{n+1}{e}^{j\Delta ti\Delta x}+{\displaystyle \frac{1}{12{(\Delta x)}^2}}{\xi }^{n+1}{e}^{j\Delta t(i+1)\Delta x}\\ \hfill +{\displaystyle \frac{1}{12{(\Delta x)}^2}}{\xi }^{n+1}{e}^{j\Delta t(i-1)\Delta x}={\displaystyle \frac{16}{12{(\Delta x)}^2}}{\xi }^n{e}^{j\Delta t(i-1)\Delta x}\\ \hfill +{\displaystyle \frac{1}{\Delta t}}{\xi }^n{e}^{j\Delta ti\Delta x}+{\displaystyle \frac{16}{12{(\Delta x)}^2}}{\xi }^n{e}^{j\Delta t(i+1)\Delta x},\end{array}$$

(71)

$$\begin{array}{c}\hfill \left({\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{30}{12{(\Delta x)}^2}}\right){\xi }^{n+1}{e}^{j\Delta ti\Delta x}+{\displaystyle \frac{5}{12{(\Delta x)}^2}}\left({\xi }^{n+1}{e}^{j\Delta t(i+1)\Delta x}+{\xi }^{n+1}{e}^{j\Delta t(i-2)\Delta x}\right)\\ \hfill ={\displaystyle \frac{1}{\Delta t}}{\xi }^n{e}^{j\Delta ti\Delta x}+{\displaystyle \frac{80}{12{(\Delta x)}^2}}\left({\xi }^n{e}^{j\Delta t(i+1)\Delta x}+{\xi }^n{e}^{j\Delta t(i-1)\Delta x}\right),\end{array}$$

(72)

Equations (71) and (72) are simplified to obtain the following

$$\begin{array}{c}\hfill \xi ={\displaystyle \frac{{\displaystyle \frac{16}{6{(\Delta x)}^2}}cos\left(\omega \right)+{\displaystyle \frac{1}{\Delta t}}}{{\displaystyle \frac{1}{\Delta t}}+1+{\displaystyle \frac{30}{12{(\Delta x)}^2}}+{\displaystyle \frac{1}{6{(\Delta x)}^2}}cos\left(\omega \right)}},\end{array}$$

(73)

and

$$\begin{array}{c}\hfill \xi ={\displaystyle \frac{{\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{80cos\left(\omega \right)}{6{(\Delta x)}^2}}}{{\displaystyle \frac{1}{\Delta t}}+{\displaystyle \frac{30}{{(\Delta x)}^2}}+{\displaystyle \frac{5}{6{(\Delta x)}^2}}cos\left(\omega \right)}},\end{array}$$

(74)

where $\omega =\Delta t\Delta x$. By using the inequality $\left|\xi \right|\le 1$, when $\omega =\pi$, we obtain the following, respectively

$$\begin{array}{c}\hfill \Delta t>{\displaystyle \frac{24}{4-12{(\Delta x)}^2}},\end{array}$$

(75)

and

$$\begin{array}{c}\hfill \Delta t>{\displaystyle \frac{6{(\Delta x)}^2}{35}},\end{array}$$

(76)

4.3.6. Consistency with the System of ADR by Using HUPFD

By discretizing (56) and (57) by using the HUPFD1 scheme, we obtain the following

$$\begin{array}{cc}\hfill {\displaystyle \frac{v_i^{n+1}-v_i^n}{\Delta t}}+c{\displaystyle \frac{v_{i+1}^{n+1}-v_i^n}{\Delta x}}& =-q_1{u}_i^{n+1}-q_2{v}_i^{n+1}\hfill \\ \hfill & +{\mu }_1\left({\displaystyle \frac{-v_{i+2}^{n+1}+16v_{i+1}^n-30v_i^{n+1}+16v_{i-1}^n-v_{i-2}^{n+1}}{12{(\Delta x)}^2}}\right),\hfill \end{array}$$

(77)

From (77), we obtain the following Taylor’s expansions

$$\begin{array}{cc}\hfill u_i^{n+1}& \approx u_i^n+\Delta t{\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+{\displaystyle \frac{{(\Delta t)}^3}{6}}{\displaystyle \frac{{\partial }^{3}u}{\partial t^3}}+\cdots \hfill \end{array}$$

(78)

$$\begin{array}{cc}\hfill u_i^{n-1}& \approx u_i^n-\Delta t{\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}-{\displaystyle \frac{{(\Delta t)}^3}{6}}{\displaystyle \frac{{\partial }^{3}u}{\partial t^3}}+\cdots \hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill u_{i-1}^n& \approx u_i^n-\Delta x{\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{{(\Delta x)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}-{\displaystyle \frac{{(\Delta x)}^3}{6}}{\displaystyle \frac{{\partial }^{3}u}{\partial x^3}}+\cdots \hfill \end{array}$$

(80)

$$\begin{array}{cc}\hfill u_{i+1}^n& \approx u_i^n+\Delta x{\displaystyle \frac{\partial u}{\partial x}}+{\displaystyle \frac{{(\Delta x)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{{(\Delta x)}^3}{6}}{\displaystyle \frac{{\partial }^{3}u}{\partial x^3}}+\cdots \hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill u_{i-2}^{n+1}& \approx u_i^n+\Delta t{\displaystyle \frac{\partial u}{\partial t}}-2\Delta x{\displaystyle \frac{\partial u}{\partial x}}+2{(\Delta x)}^2{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}-2(\Delta t\Delta x){\displaystyle \frac{{\partial }^{2}u}{\partial t\partial x}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+\cdots \hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill u_{i+2}^{n+1}& \approx u_i^n+\Delta t{\displaystyle \frac{\partial u}{\partial t}}+2\Delta x{\displaystyle \frac{\partial u}{\partial x}}+2{(\Delta x)}^2{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+2(\Delta t\Delta x){\displaystyle \frac{{\partial }^{2}u}{\partial t\partial x}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+\cdots \hfill \end{array}$$

(83)

and

$$\begin{array}{cc}\hfill v_i^{n+1}& \approx v_i^n+\Delta t{\displaystyle \frac{\partial v}{\partial t}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}v}{\partial t^2}}+{\displaystyle \frac{{(\Delta t)}^3}{6}}{\displaystyle \frac{{\partial }^{3}v}{\partial t^3}}+\cdots \hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill v_i^{n-1}& \approx v_i^n-\Delta t{\displaystyle \frac{\partial v}{\partial t}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}v}{\partial t^2}}-{\displaystyle \frac{{(\Delta t)}^3}{6}}{\displaystyle \frac{{\partial }^{3}v}{\partial t^3}}+\cdots \hfill \end{array}$$

(85)

$$\begin{array}{cc}\hfill v_{i-1}^n& \approx v_i^n-\Delta x{\displaystyle \frac{\partial v}{\partial x}}+{\displaystyle \frac{{(\Delta x)}^2}{2}}{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}-{\displaystyle \frac{{(\Delta x)}^3}{6}}{\displaystyle \frac{{\partial }^{3}v}{\partial x^3}}+\cdots \hfill \end{array}$$

(86)

$$\begin{array}{cc}\hfill v_{i+1}^n& \approx v_i^n+\Delta x{\displaystyle \frac{\partial v}{\partial x}}+{\displaystyle \frac{{(\Delta x)}^2}{2}}{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+{\displaystyle \frac{{(\Delta x)}^3}{6}}{\displaystyle \frac{{\partial }^{3}v}{\partial x^3}}+\cdots \hfill \end{array}$$

(87)

$$\begin{array}{cc}\hfill v_{i-2}^{n+1}& \approx v_i^n+\Delta t{\displaystyle \frac{\partial v}{\partial t}}-2\Delta x{\displaystyle \frac{\partial v}{\partial x}}+2{(\Delta x)}^2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}-2(\Delta t\Delta x){\displaystyle \frac{{\partial }^{2}v}{\partial t\partial x}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}v}{\partial t^2}}+\cdots \hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill v_{i+2}^{n+1}& \approx v_i^n+\Delta t{\displaystyle \frac{\partial v}{\partial t}}+2\Delta x{\displaystyle \frac{\partial v}{\partial x}}+2{(\Delta x)}^2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+2(\Delta t\Delta x){\displaystyle \frac{{\partial }^{2}v}{\partial t\partial x}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}v}{\partial t^2}}+\cdots \hfill \end{array}$$

(89)

By applying Taylor’s expansions to (77), respectively, we obtain the following simplified equations

$$\begin{array}{cc}\hfill & (1-q_1)u_i^n+\left(1+\Delta t-q_1\Delta t+{\displaystyle \frac{32{\mu }_2\Delta t}{12{(\Delta x)}^2}}+{\displaystyle \frac{c\Delta t}{12}}\right){\displaystyle \frac{\partial u}{\partial t}}\hfill \\ \hfill & +c{\displaystyle \frac{\partial u}{\partial x}}+\left({\displaystyle \frac{\Delta t}{2}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}-{\displaystyle \frac{q_1{(\Delta t)}^2}{2}}+{\displaystyle \frac{17{\mu }_1{(\Delta t)}^2}{12{(\Delta x)}^2}}+{\displaystyle \frac{c{(\Delta t)}^2}{2\Delta x}}\right){\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\hfill \\ \hfill & +\left({\displaystyle \frac{c(\Delta x)}{2}}-{\mu }_1\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}+{\displaystyle \frac{c\Delta t}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t\partial x}}-q_2{v}_i^n=0,\hfill \end{array}$$

(90)

and

$$\begin{array}{c}{q}_2{v}_i^n+\left(1+q_2\Delta t+{\displaystyle \frac{32\Delta t{\mu }_2}{12{(\Delta x)}^2}}+{\displaystyle \frac{c{(\Delta t)}^2}{\Delta x}}\right){\displaystyle \frac{\partial v}{\partial t}}+c{\displaystyle \frac{\partial v}{\partial x}}\hfill \\ \hfill +\left(\Delta t+q_2{(\Delta t)}^2+{\displaystyle \frac{32{\mu }_2{(\Delta t)}^2}{12{(\Delta )}^2}}+{\displaystyle \frac{c{(\Delta )}^2}{2(\Delta x)}}\right){\displaystyle \frac{{\partial }^{2}v}{\partial t^2}}+\left({\displaystyle \frac{c(\Delta x)}{2}}{\displaystyle \frac{4{\mu }_2}{12}}-{\displaystyle \frac{12{\mu }_2}{12}}\right){\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}\\ \hfill +q_1\left(u_i^n+\Delta t{\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{{(\Delta t)}^2}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\cdots \cdots \right)=0,\end{array}$$

(91)

Let $\Delta t={(\Delta x)}^3$, which leads to Equations (90) and (91) being written as follows

$$\begin{array}{cc}\hfill & (1-q_1)u_i^n+\left(1+{(\Delta x)}^3-q_1{(\Delta x)}^3+{\displaystyle \frac{32{\mu }_1\Delta x}{12}}+c{(\Delta x)}^2\right){\displaystyle \frac{\partial u}{\partial t}}+c{\displaystyle \frac{\partial u}{\partial x}}\hfill \\ \hfill & +\left({\displaystyle \frac{{(\Delta x)}^3}{2}}+{\displaystyle \frac{{(\Delta x)}^6}{2}}-{\displaystyle \frac{q_1{(\Delta x)}^6}{2}}+{\displaystyle \frac{17{\mu }_1{(\Delta x)}^4}{12}}+{\displaystyle \frac{c{(\Delta x)}^5}{2}}\right){\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}+\left({\displaystyle \frac{c(\Delta x)}{2}}-{\mu }_1\right){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}\hfill \\ \hfill & +{\displaystyle \frac{c{(\Delta x)}^3}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t\partial x}}-q_2{v}_i^n=0,\hfill \end{array}$$

(92)

and

$$\begin{array}{cc}\hfill & q_2{v}_i^n+\left(1+q_2{(\Delta x)}^3+{\displaystyle \frac{32{\mu }_2\Delta x}{12}}+c{(\Delta x)}^5\right){\displaystyle \frac{\partial v}{\partial t}}+c{\displaystyle \frac{\partial v}{\partial x}}\hfill \\ \hfill & +\left({(\Delta x)}^3+q_2{(\Delta x)}^6+{\displaystyle \frac{32{\mu }_2{(\Delta x)}^4}{12}}+{\displaystyle \frac{c{(\Delta x)}^5}{2}}\right){\displaystyle \frac{{\partial }^{2}v}{\partial t^2}}\hfill \\ \hfill & +\left({\displaystyle \frac{c(\Delta x)}{2}}-{\mu }_2\right){\displaystyle \frac{{\partial }^{2}v}{\partial x^2}}+q_1\left(u_i^n+{(\Delta x)}^3{\displaystyle \frac{\partial u}{\partial t}}+{\displaystyle \frac{{(\Delta x)}^6}{2}}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}}\right)=0,\hfill \end{array}$$

(93)

When $\Delta x⟶0$, we obtain the original system ADR equations

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial u}{\partial t}}+c{\displaystyle \frac{\partial u}{\partial x}}& =(q_1-1)u+(q_2+s)v+{\mu }_1{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}},\hfill \end{array}$$

(94)

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial v}{\partial t}}+c{\displaystyle \frac{\partial v}{\partial x}}& =-q_{1}u-q_{2}v+{\mu }_2{\displaystyle \frac{{\partial }^{2}v}{\partial x^2}},\hfill \end{array}$$

(95)

Theorem 1

(Solution positivity). Let $u^0\left(x\right)\ge 0$, $0\le f^0\left(x\right)\le 1$ and $v^0\left(x\right)\ge 0$ for the numerical scheme (11), (20) and (60); we have $u_i^{n+1}\ge 0$, $0\le f_i^{n+1}$ and $v_i^{n+1}\ge 0$, independent of the mesh step size [31].

Proof. 

Let us assume that $u_i^{n+1}\ge 0$, $0\le f_i^{n+1}\le 1$ and $v_i^n\ge 0$, for $i=0,1,\cdots \cdots ,N$, $n=0,\cdots \cdots N$.

Equations (11), (20) and (60) comprise a coefficient matrix with a strictly diagonally dominant tridiagonal matrix, with positive values on the diagonal and sub-diagonals, which implies Equations (11), (20) and (60) yield non-negative inverses. Thus, the solution to the algebraic system is non-negative in the form of $u_i^{n+1}\ge 0$ and $v_i^{n+1}\ge 0$, for $i=0,1,\cdots \cdots N$.

Furthermore, it is observed that $f_i^{n+1}\ge 0$ in equations (6), (15), (54) and (55). By applying the same consideration at each time level, we complete the proof. □

5. Formulation of the POD Technique

In this section, the POD basis function is constructed and used to reduce the degree of freedom to the UPFD and HUPFD scheme.

The POD technique is used for the HUPFD by extracting the snapshot matrix in order to produce the EHUPFD, which leads to the construction of the approximate state variable ${\psi }_i$ by developing a small subset of the leading basis function obtained from the SVD factorization. The POD technique seeks a low-dimensional basis function that can accurately approximate the state variable, which is a left singular vector from the SVD.

The first L sequence of solutions is extracted from the HUPFD solutions as given by $u_i^n$, $v_i^n$ and $s_i^n$, which gives the formulated $n\times L$ snapshot matrix given below

$$\begin{array}{c}\hfill A=\left(\begin{array}{ccccccccc}{u}_1^1& u_1^2& ·& ·& ·& ·& ·& ·& u_1^L\\ u_2^1& u_2^2& ·& ·& ·& ·& ·& ·& u_2^L\\ ·& ·& & & & & & & ·\\ ·& ·& ·& & & & & & ·\\ ·& ·& & ·& & & & & ·\\ ·& ·& & & ·& & & & ·\\ ·& ·& & & & ·& & & ·\\ ·& ·& & & & & ·& & ·\\ ·& ·& & & & & & ·& ·\\ u_m^1& u_m^2& ·& ·& ·& ·& ·& ·& u_m^L\end{array}\right),\end{array}$$

(96)

where m and L represent the size of the snapshot matrix, which is factorized by using the singular value decomposition (SVD) as given below

$$\begin{array}{c}\hfill A=U\Sigma V^T=U\left(\begin{array}{cc}{\Sigma }_{l\times l}& O_{l\times (L-l)}\\ O_{(m-1)\times l}& O_{(m-1)\times (L-l)}\end{array}\right)V^T,\end{array}$$

(97)

where $U=({\psi }_1,{\psi }_2,\dots ,{\psi }_m)$ is an $m\times m$ matrix representing an orthogonal matrix consisting of the eigenvectors of $A{A}^T$, ${\Sigma }_{l\times l}=$diag$({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_l)$ represents the diagonal matrix from the SVD, given in decreasing order ${\sigma }_1\ge {\sigma }_2\ge \dots \ge {\sigma }_l>0$, and $V=({\varphi }_1,{\varphi }_2,\dots ,{\varphi }_L)$ is an $L\times L$ orthogonal matrix which represent the orthogonal eigenvector of $A^{T}A$ and $l=rankA$; O is a zero matrix [32].

From identical eigenvalues of $A^{T}A$ and $A{A}^T$, we obtain the eigenvalues of ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_l>0(l=rankA)$ for matrices $A{A}^T$ and corresponding eigenvectors ${\psi }_j$, which is given by the formula below

$$\begin{array}{c}\hfill {\psi }_j=A{\varphi }_j/\sqrt{{\lambda }_j},j=1,2,\cdots \cdots ,l,\end{array}$$

(98)

This leads to the construction of the matrix $A_M$ given below

$$\begin{array}{c}\hfill A_M=U\Sigma V^T=U\left(\begin{array}{cc}{\Sigma }_{M\times M}& O_{M\times (L-M)}\\ O_{(m-M)\times l}& O_{(m-M)\times (L-M)}\end{array}\right)V^T,\end{array}$$

(99)

where the diagonal matrix given by ${\Sigma }_{M\times M}=diag({\sigma }_1,{\sigma }_1,\cdots ,{\sigma }_m)$ is constituted of the first M positive singular values of the diagonal matrix.

5.1. The Implementation of the Algorithm of the EHUPFD Scheme for the Advection Diffusion Reaction Equations

Below are the steps to implement the algorithm of the EHUPFD to solve the linear, non-linear and system ADR equations [32].

Step 1:

Snapshot matrix for HUPFD is extracted.

Step 2:

Formulation of the matrix $A={\left(u_i^n\right)}_{m\times L}$ and calculation of eigenvalues and eigenvectors of matrix $A^{T}A$ and $A{A}^T$ executed.

Step 3:

Choice of POD basis for the error tolerance of $\mu =O(\Delta t,\Delta x^2)$ is executed.

Step 4:

The EHUPFD computes the reduced-order solution.

Step 5:

Accuracy of the solution is checked for the renewal of the POD basis.

5.2. The EHUPFD Error Estimate

The following equation is used to estimate the error of the EHUPFD when solving the ADR equations.

$$\begin{array}{c}\hfill {\parallel u^n-u^{*n}\parallel }_2={\parallel u^n-u_M^n\parallel }_2={\parallel (A-\varphi {\varphi }^{T}A){\xi }_n\parallel }_2\le \sqrt{{\lambda }_{M+1}},\end{array}$$

(100)

$n=1,2,\cdots \cdots ,L$

Theorem 2

(Accuracy). The error estimate for the HUPFD $u_i^n$ and $u_i^{*n}$ from the EHUPFD is calculated by using the following equation

$$\begin{array}{c}\hfill |u_i^n-u_i^{*n}{|\le (1+\delta )}^{n-L}\sqrt{{\lambda }_M+1}\mathit{for}(L+1\le n\le N),\end{array}$$

(101)

$$\begin{array}{c}\hfill \mathit{where}\delta ={\displaystyle \frac{\Delta t}{\Delta x^2}}{\parallel B\parallel }_{2,2}.\end{array}$$

(102)

The absolute error, calculated between the actual and approximate solutions from the POD reduced-order model satisfy the following equation [32]

$$\begin{array}{c}\hfill |u(x_i,t_n)-u_i^{*n}|=O\left({(1+\delta )}^{n-L}\sqrt{{\lambda }_M+1},\Delta t,\Delta x^2\right),1\le i\le M,1\le n\le N,\end{array}$$

(103)

6. Results and Discussion

In this section, we present solutions to the linear, non-linear and system ADR equations solved by using the EHUPFD, UPFD, NSFD and Crank–Nicolson methods. To check the performance of the methods, absolute errors, convergence rates and computational time are measured. The actual and numerical solutions are denoted by $u_{i,j}$ and ${\overline{u}}_{i,j}$, respectively. The error is represented by $e_{i,j}$ and its magnitude is calculated by using norm to infinity denoted by $L_{\infty }$-norm given by the formula below,

$$\begin{array}{c}\hfill \left|\right|e_{i,n}{\left|\right|}_{\infty }=\left|\right|u_{i,n}-{\overline{u}}_{i,n}{\left|\right|}_{\infty }=Max|u_{i,n}-{\overline{u}}_{i,n}|,\end{array}$$

(104)

The formula to calculate the convergence rate is given below,

$$\begin{array}{c}\hfill q(\Delta t_k)={\displaystyle \frac{lo{g}_2\left(\frac{e_k}{e_{k+1}}\right)}{lo{g}_2\left(\frac{\Delta t}{\Delta t_{k+1}}\right)}}\mathrm{or}q(\Delta x_k)={\displaystyle \frac{lo{g}_2\left(\frac{e_k}{e_{k+1}}\right)}{lo{g}_2\left(\frac{\Delta x}{\Delta x_{k+1}}\right)}},\end{array}$$

(105)

Finding the best snapshot size is paramount because it influences the EHUPFD solution accuracy. The best snapshot is determined by finding the solution to the problem when $L=1$ and then an increment of 1 to L is evaluated until there is no significant change in the solution. The tolerance is calculated using the formula below

$$\left|\right|u_{exact}-u_L{\left|\right|}_{\infty }\le ϵ,L=1,2,3,\dots$$

(106)

where $u_L$ is a solution for a specific value of L and $u_{exact}$ is the exact solution. The first value of L that satisfies Equation (106) is the optimal value of L. We set $ϵ={10}^{-15}$.

6.1. Results Obtained to Solve the Linear ADR Equation

The convergence rates of the UPFD, Crank–Nicolson, NSFD and HUPFD with respect to the time and spatial steps are shown in

Table 1. Convergence rate results obtained from the UPFD, NSFD, Crank–Nicolson and HUPFD methods when solving the ADR equation.

		q(Δtk)
Δtk	Δxk	UPFD	HUPFD	EHUPFD	Crank–Nicolson	NSFD
$0.00005$	$0.6667$	$2.1675$	$2.5233$	$2.5233$	$4.5606$	$2.0579$
$0.0001$	$0.6667$	$0.9973$	$1.0054$	$1.0054$	$1.1996$	$1.9427$
$0.0002$	$0.6667$	$1.8201$	$1.9990$	$1.9990$	$2.7372$	$1.8871$
$0.0004$	$0.6667$

. The results show that the EHUPFD converges to the solution faster than the UPFD.

Table 2. Numerical results obtained when solving the linear ADR equation by using the UPFD, HUPFD, EHUPFD, NSFD and Crank–Nicolson methods.

(x, t)	Exact Solution	UPFD Solution	Absolute Error	EHUPFD	Absolute Error	Crank–Nicolson	Absolute Error	NSFD	Absolute Error
$(1.3334,0.0002)$	$0.5136$	$0.5137$	$9.3425\times {10}^{-5}$	$0.5125$	$0.0011$	$0.5138$	$1.6145\times {10}^{-4}$	$0.5137$	$9.4142\times {10}^{-5}$
$(2.6668,0.0004)$	$0.1355$	$0.1356$	$8.1664\times {10}^{-5}$	$0.1366$	$0.0011$	$0.1356$	$6.7682\times {10}^{-5}$	$0.1356$	$7.4518\times {10}^{-5}$
$(4.0002,0.0006)$	$0.0357$	$0.0358$	$5.1090\times {10}^{-4}$	$0.0363$	$3.2521\times {10}^{-5}$	$0.0358$	$2.9601\times {10}^{-5}$	$0.0358$	$3.2769\times {10}^{-5}$

compares solutions obtained by the exact, UPFD, Crank–Nicolson, NSFD and EHUPFD solutions at $(t,x)=\left(1.33334,0.0002\right),\left(2.6668,0.0004\right)$ and $\left(4.0002,0.0006\right)$.

Table 3. Infinity norm results obtained by using the exact, UPFD, HUPFD, EHUPFD, Crank–Nicolson and NSFD methods to solve the linear ADR equation.

(x, t)	Exact	UPFD	Error	HUPFD	Error	EHUPFD	Error	Crank–Nicolson	Error	NSFD	Error
${∥u(:,0.0002)∥}_{\infty }$	$1.1654$	$1.1654$	$0.0025$	$1.1654$	$0.0025$	$1.1655$	$0.0025$	$1.1654$	$0.0025$	$1.1654$	$0.0025$
${∥u(:,0.0004)∥}_{\infty }$	$1.1657$	$1.1657$	$0.0025$	$1.1657$	$0.0025$	$1.1657$	$0.0025$	$1.1657$	$0.0025$	$1.1657$	$0.0025$
${∥u(:,0.0006)∥}_{\infty }$	$1.1659$	$1.1660$	$0.0025$	$1.1660$	$0.0025$	$1.1660$	$0.0025$	$1.1660$	$0.0025$	$1.1657$	$0.0025$
Time		$0.001230$		$0.002748$		$0.002717$		$0.003089$		$0.002689$

compares the infinity norm results of the developed EHUPFD method against the Crank Nicolson and NSFD, as well as their computational times. The results show that the EHUPFD takes less computational time to converge to the solution as compared to the HUPFD methods.

Table 3 compares the infinity norms of the proposed EHUPFD methods with the UPFD, Crank–Nicolson and NSFD methods in solving the linear ADR equation. The results show that EHUPFD methods preserve solution accuracy.

Figure 1 shows the contribution made by each POD basis mode. The first five modes contain the predominant information where the EHUPFD meets the theoretical accuracy requirement. We observed that five modes are significant in optimizing the linear ADR equation. Figure 2b compares the number of POD modes against the absolute error. We observe that only five POD modes are required in order to converge to the toleration of ${10}^{-14}$.

Figure 3a to Figure 4a show the surface plots of the exact, UPFD and EUPFD solutions of the linear ADR. Figure 4b shows the increase in error when the POD modes increases. The figures show that the developed methods achieve positivity of the solutions and the results are comparable with well-known methods such as Crank–Nicolson and NSFD.

6.2. Results Obtained by Using the Non-Linear ADR Equation

The convergence rates of the UPFD, Crank–Nicolson, NSFD and the EHUPFD with respect to the time and spatial steps are shown in

Table 4. Convergence rate results obtained when solving the non-linear ADR reaction equation by the UPFD, HUPFD, Crank–Nicolson, NSFD and EHUPFD methods.

q(Δtk)
Δ${\mathit{t}}_{\mathit{k}}$	Δ${\mathit{x}}_{\mathit{k}}$	UPFD	HUPFD	EHUPFD	Crank–Nicolson	NSFD
$0.0005$	$0.6667$	$1.7166$	$1.7197$	$1.7197$	$1.7197$	$1.71652$
$0.001$	$0.6667$	$1.7160$	$1.7150$	$1.7151$	$1.7150$	$1.71594$
$0.002$	$0.6667$	$1.7168$	$1.7195$	$1.7195$	$1.7195$	$1.71671$
$0.004$	$0.6667$

.

Table 5. Solutions obtained by the exact, UPFD, HUPFD and EHUPFD methods when solving the non-linear ADR equation.

(x, t)	Exact Solution	UPFD Solution	Absolute Error	HUPFD Solution	Absolute Error	EHUPFD Solution	Absolute Error
$(0.03338,0.002)$	$6.5475\times {10}^{-9}$	$5.4779\times {10}^{-9}$	$1.0697\times {10}^{-9}$	$5.4763\times {10}^{-9}$	$1.0712\times {10}^{-9}$	$5.4908\times {10}^{-9}$	$1.0567\times {10}^{-9}$
$(0.0676,0.004)$	$8.1536\times {10}^{-8}$	$6.8787\times {10}^{-8}$	$1.2749\times {10}^{-8}$	$6.8751\times {10}^{-8}$	$1.2784\times {10}^{-8}$	$6.9355\times {10}^{-8}$	$1.2181\times {10}^{-8}$
$(0.1014,0.006)$	$8.4478\times {10}^{-7}$	$7.1808\times {10}^{-7}$	$1.2670\times {10}^{-7}$	$7.1759\times {10}^{-7}$	$1.2718\times {10}^{-7}$	$7.2659\times {10}^{-7}$	$1.1819\times {10}^{-7}$

compares the exact, UPFD and EHUPFD solutions at $(t,x)=\left(0.03338,0.002\right),\left(0.0676,0.004\right)$ and $\left(0.1014,0.006\right)$. The HUPFD has a slightly higher convergence rate than UPFD, while the EHUPFD has significantly reduced the computational time taken to converge to the solution of the ADR equation.

In

Table 6. Infinity norm results obtained when solving the non-linear ADR equation by using the exact, UPFD, NSFD and HUPFD methods.

(x, t)	Exact	UPFD	Error	HUPFD	Error	EHUPFD	Error	Crank–Nicolson	Error	NSFD	Error
${∥u(:,0.002)∥}_{\infty }$	$2.0590$	$2.0127$	$0.0512$	$2.0127$	$0.0512$	$2.0126$	$0.0513$	$2.0127$	$0.0512$	$2.0127$	$0.0512$
${∥u(:,0.004)∥}_{\infty }$	$2.0591$	$2.0127$	$0.0512$	$2.0126$	$0.0514$	$2.0123$	$0.0514$	$2.0126$	$0.0514$	$2.0127$	$0.0512$
${∥u(:,0.004)∥}_{\infty }$	$2.0592$	$2.0128$	$0.0512$	$2.0126$	$0.0515$	$2.0121$	$0.0515$	$2.0126$	$0.0515$	$2.0128$	$0.0512$
Time		$0.001227$		$0.004136$		$0.000492$		$0.004205$		$0.003954$

, we compare the infinity norms of the proposed HUPFD with the UPFD, Crank–Nicolson and NSFD methods in solving the linear ADR equation. For this example, the results show that the EHUPFD method preserves the accuracy and reduces the computational time taken to converge to the solution of the ADR equation.

Figure 5a shows the contribution made by each POD basis mode. The first five modes contain the predominant information where the EHUPFD meets the theoretical accuracy requirement. We observed five modes are significant in optimizing the linear ADR equation. Figure 5b shows the increase in computational time when the number of modes increases.

Figure 6a compares the results obtained by the EHUPFD, Crank–Nicolson and NSFD methods. The results show that the EHUPFD solution is comparable with the Crank–Nicolson and NSFD solutions.

Figure 6b compares the error against the number of modes. The results show that the error of the solution decreases when the number of modes increases.

Figure 7a to Figure 8a show the surface plots of the exact, UPFD and EUPFD solutions of the non-linear ADR equation. Figure 8b show the absolute errors between the exact solution and the EHUPFD solution for the non-linear ADR against time.

6.3. Results Obtained When Solving the Schnakenberg Model

The convergence rates of the UPFD, HUPFD and EHUPFD with respect to the time and spatial steps are shown in

Table 7. Convergence rate when solving the Schnakenberg model by using the UPFD, HUPFD and EHUPFD methods.

	q(Δtk)
Δtk	Δxk	UPFD	HUPFD	EHUPFD
$0.00005$	$0.6667$	$1.0002$	$1.0029$	$1.0027$
$0.0001$	$0.6667$	$1.0004$	$1.0058$	$1.0055$
$0.0002$	$0.6667$	$1.0007$	$1.0116$	$1.0109$
$0.0004$	$0.6667$

. The results show that the convergence rate of the EHUPFD methods is slightly higher than the UPFD when solving the Schnakenberg model.

Table 8. Solutions u obtained by the UPFD, HUPFD and EHUPFD methods when solving the Schnakenberg system equation.

(x, t)	UPFD	HUPFD	EHUPFD
$u(:,0.0002)$	$0.0048$	$0.0050$	$0.0050$
$u(:,0.0004)$	$0.0182$	$0.0198$	$0.0198$
$u(:,0.0006)$	$0.0690$	$0.0788$	$0.0788$
Time	$0.005959$	$0.024321$	$0.004894$

and

Table 9. Solutions v obtained by the UPFD, HUPFD and EHUPFD methods when solving the Schnakenberg system equation.

(x, t)	UPFD	HUPFD	EHUPFD
$u(:,0.0002)$	$0.0048$	$0.5310$	$0.5310$
$u(:,0.0004)$	$0.0186$	$0.1429$	$0.1429$
$u(:,0.0006)$	$0.0711$	$0.0390$	$0.0390$
Time	$0.005312$	$0.021700$	$0.004220$

compare the solutions of the UPFD, HUPFD and EHUPFD solutions at $(t,x)=\left(1.33334,0.0002\right),\left(2.6668,0.0004\right)$ and $\left(4.0002,0.0006\right)$, as well as the computational times. The results show that the EHUPFD takes less computational time to converge to the solution than the UPFD, HUPFD and EHUPFD methods. Figure 9a to Figure 10b show the surface plots of the UPFD, HUPFD and EHUPFD of the Schnakenberg model. The figures show that the developed methods achieve positivity of the solutions. Figure 11 compares the solutions obtained by the exact HUPFD, EHUPFD and UPFD. The results show that the proposed methods are comparable to the known methods of the UPFD.

The convergence rates of the UPFD, HUPFD and EHUPFD with respect to the time and spatial steps are shown in Table 7. The results show that the convergence rate of the EHUPFD methods is slightly higher than the UPFD when solving the Schnakenberg model. Table 8 and Table 9 compare the UPFD, HUPFD and EHUPFD solutions at $(x,t)=\left(1.33334,0.0002\right),\left(2.6668,0.0004\right)$ and $\left(4.0002,0.0006\right)$, as well as the computational times. The results show that the EHUPFD takes less computational time to converge to the solution than the UPFD, HUPFD and EHUPFD methods. Figure 9a to Figure 10b show the surface plots of the UPFD, HUPFD and EHUPFD of the Schnakenberg model. The figures show that the developed methods achieve positivity of the solutions. Figure 11 compares the solutions obtained by the exact HUPFD, EHUPFD and UPFD. The results show that the proposed methods are comparable to the know methods of the UPFD.

7. Conclusions

This paper uses the EHUPFD to find solutions to the linear, non-linear and system advection diffusion reaction equations. The goal was to investigate the performance of the EHUPFD in terms of computational time, absolute error and convergence rate. The results show that the EHUPFD gives solutions that are a good approximation to solve the linear, non-linear and system ADR equations, which validates the method’s efficiency because it confirms the theoretical results. When solving the ADR equations by EHUPFD, a few modes containing the predominant information are needed to satisfy the theoretical results. The theoretical results are achieved using only 4 POD bases from the HUPFD’s 40 classical solutions as a snapshot. This implies that the EHUPFD saves computational time, reduces the degree of freedom and alleviates truncation error. The EHUPFD method also preserves the positivity of the solutions.