# Highly Accurate and Efficient Time Integration Methods with Unconditional Stability and Flexible Numerical Dissipation

^{1}

^{2}

^{*}

## Abstract

**:**

^{m}algorithm and the method of storing an incremental matrix are employed in the calculation of the generalized Padé approximations. The proposed methods can achieve higher-order accuracy, unconditional stability, flexible dissipation, and zero-order overshoots. For linear transient problems, the accuracy of the proposed methods can reach 10

^{−16}(computer precision), and they enjoy advantages both in accuracy and efficiency compared with some well-known explicit Runge–Kutta methods, linear multi-step methods, and composite methods in solving nonlinear problems.

## 1. Introduction

_{∞}-Bathe method [10], the Kim method [11], and the TR-TR-BIF [12]. Among these methods, the TR-TR-BIF [12] proposed by the present authors has been generalized to the dynamic analysis of multibody systems [14] and structures under seismic response [15,16], further showing its superiority in the analysis of transient problems. Recently, composite methods with higher-order accuracy have been constructed [17,18,19], and they show a considerable advantage in phase accuracy compared with the second-order accurate composite methods.

^{m}algorithm, and the technology of storing incremental matrices is used in the calculation of responses of linear parts. The responses of nonlinear parts are approximated by the combination of the Gauss–Legendre quadrature formula and the explicit Runge–Kutta method. The two combinations can ensure that the proposed methods can accurately and quickly compute the responses of transient problems. Numerical experiments validate that when the proposed methods have the same computation as other methods, the accuracy of the proposed methods is greater than or equal to three orders of magnitude. The theoretical analysis finds that the time integration methods can obtain higher-order accuracy, unconditional stability, controllable dissipation, and zero-order overshoots. Therefore, the proposed methods are suitable both for conservative and non-conservative systems due to flexibly controllable numerical properties.

## 2. Basic Idea of the New Strategy

**H**is a matrix that includes eigenvalues with large negative real parts or with purely imaginary eigenvalues of large modulus [47], and the nonlinear term

**f**is supposed to be a non-stiff satisfying the Lipschitz condition. If the term

**f**only relates to time t, the nonlinear Equation (1) reduces to the linear one. The discretized dynamic system (1) arises in many applications [37,38,39,40,41,42,43,44,45,46,47,49,50,51], such as structural dynamics [39,40,41], multibody dynamics [45], molecular dynamics [49], and so on. It is well-known that the general solution [47] of Equation (1) has the form of

**A**

_{ana}= e

^{ΔtH}can transfer the free responses of the previous step, and the forced responses of the current step are computed by the analytical vector

**L**

_{ana}= ${{\displaystyle \int}}_{t}^{t+\u2206t}{e}^{\u2206t\mathit{H}}\mathit{f}(\mathit{y},\tau )d\tau $. For practical systems, especially for large-scale problems, the computations of the matrix

**A**

_{ana}and the vector

**L**

_{ana}are expensive. The task of this strategy is to quickly construct highly accurate substitutes for

**A**

_{ana}and

**L**

_{ana}.

**H**) is used to approximate

**A**

_{ana}, and the corresponding numerical matrix

**A**

_{num}can be formulated as

_{∞}< 1, it has (2n − 1)th-order accuracy and (2n)th-order accuracy if ρ

_{∞}= 1. To further improve the accuracy of

**A**

_{num}(∆t

**H**) given in Equation (3), the 2

^{m}algorithm and storage of incremental matrix technology are used in the preparation of

**A**

_{num}(∆t

**H**), which is shown below. Applying Equation (5) to Equation (3) can yield

**A**

_{num}(∆t

**H**) = $\stackrel{\_}{\mathit{Q}}$(∆t

**H**)

^{−1}$\stackrel{\_}{\mathit{P}}$(∆t

**H**) that is equivalent to $\stackrel{\_}{\mathit{Q}}$(∆t

**H**)

**A**

_{num}(∆t

**H**) = $\stackrel{\_}{\mathit{P}}$(∆t

**H**) can be reformulated as

**I**is defined as ∆

**S**(∆t

**H**), as follows:

**A**

_{num}(∆t

**H**). Here, a time step size ∆t is divided into N = 2

^{m}parts, leading to

**S**(∆t

_{N}

**H**) is very small compared with the identity matrix

**I**, so during the calculation of Equation (10), the ∆

**S**(∆t

_{N}

**H**) is stored instead of

**A**

_{num}(∆t

_{N}

**H**) to reduce rounding errors. It is well-known that

**A**

_{num}(∆t

**H**), Equation (11) should be iterated m times. Then, the calculation in Equation (10) is equivalent to executing the following statement

**S**(∆t

_{N}

**H**) is no longer a very small matrix, and the above addition will have no serious numerical round-off error again. To show the accuracy advantage of the method of storing incremental matrix, a simple model is considered here, in which

**H**= 1 and ∆t = 0.1. Table 1 provides absolute errors of the method of storing the total matrix and the method of storing the incremental matrix, and one can see that with the increase of m, (a) the former’s accuracy firstly increases and then continuously decreases; (b) the errors of the latter trend to zero.

**A**

_{num}(∆t

**H**) for exactly controlling the amount of numerical dissipation via ρ

_{∞}, the matrix

**A**

_{num}(∆t

**H**) shown in Equation (3) for the case of N ≥ 1 is reformulated as

**S**(∆t

**H**) in Equation (9) becomes the function of $\sqrt[N]{{\rho}_{\infty}}$.

**L**

_{ana}in Equation (2) is relatively expensive. Hence, in our work, the vector

**L**

_{ana}is approximated by r-node Gauss–Legendre quadrature method (r = 1, 2, …), and its expression has the form as

**y**at the quadrature points of t + (1 + ξ

_{l})∆t/2 (l = 1, 2, …, r), which are used in Equation (14), are calculated by the explicit Runge–Kutta methods [3], as follows

**A**

_{num}(∆t

**H**) and

**L**

_{num}(∆t

**H**), and the Newton iteration method can be avoided.

#### 2.1. Second-Order Accurate Scheme

**S**(∆t

**H**) as follows

**S**(∆t

**H**), the highly accurate matrix

**A**

_{num}(∆t

**H**), can be obtained from Equations (10)–(13). Together with Equation (23) and the Gauss–Legendre quadrature method, the transient response can be solved for linear systems. For nonlinear systems, here we adopt the modified Euler method to explicitly calculate the values of nonlinear terms

**f**(

**y**, t) at Gauss–Legendre quadrature points. The tableau of the modified Euler method has the form as

#### 2.2. Fourth-Order Accurate Scheme

**S**(∆t

**H**) of the fourth-order scheme, as follows:

_{∞}= 1, Equation (31) turns into

_{∞}= 0, Equation (31) becomes

_{∞}= 1 are the lowest. The classical fourth-order Runge–Kutta method [3] is utilized in the present scheme to compute nonlinear function

**f**(

**y**, t) at Gauss–Legendre quadrature points to avoid the loss-of-accuracy order and the tableau of the Runge–Kutta method is

## 3. Numerical Properties Analysis

_{cr}, and ensures that the constructed matrix

**A**

_{num}(∆t

**H**) can be calculated with up to computer precision, which is also discussed below.

**A**

_{num}is

_{1}= tr(

**A**

_{num}) and A

_{2}= det(

**A**

_{num}), and the two eigenvalues can be written as the form of λ

_{1,2}= a ± ib, in which i = $\sqrt{-1}$. The definition of the spectral radius [1] is

_{∞}= 1) can keep all information of a dynamic system, while the dissipative AEDMn (0 ≤ ρ

_{∞}< 1) can filter out the high-frequency modes.

#### 3.1. Stability, Dissipation, and Accuracy of the Second-Order Scheme

#### 3.1.1. Spectral Characteristics

_{∞}≤ 1, and the amount of its high-frequency dissipation can be exactly controlled via ρ

_{∞}. (b) Additionally, with the increase of m, the low-frequency range, where the spectral radius trends to 1, becomes wider and preserves more low-frequency modes.

_{∞}= 0) and AECM1 (ρ

_{∞}= 1), which are the representative schemes, are considered below for analyzing the amplitude and phase accuracy of the present scheme. Amplitude and period errors of the AEDM1 (ρ

_{∞}= 0) and AECM1 (ρ

_{∞}= 1) versus τ for the undamped case are shown in Figure 2 and Figure 3, respectively, in which it can be seen that (a) with the increase in m, both the amplitude and period errors can be simultaneously decreased; (b) for the same m, the non-dissipative scheme and the dissipative schemes almost have the same phase accuracy, implying that the numerical dissipation mainly affects the amplitude accuracy of the AEC/DM1.

_{exact}) of the AEC/DM1 for the damped case (ξ = 0.5) are shown in Figure 4, Figure 5 and Figure 6, in which ρ

_{exact}= exp(−ξτ) [52]. It follows that: (a) With the increase of m, the numerical spectral radius approaches the analytical one; (b) Among the low-frequency range, the accuracy of the AECM1 is higher than that of the AEDM1; (c) For smaller m, because the AECM1 cannot provide numerical dissipation, their spectral radii do not agree well with analytical one in the high-frequency range (τ > 10).

#### 3.1.2. Rounding Errors

_{cr}. When m < m

_{cr}, the truncation error dominates, and increasing m can improve accuracy, but when m ≥ m

_{cr}, the rounding error dominates, and increasing m cannot improve accuracy further. From the phase accuracy analysis shown in Figure 2 and Figure 3, one can find that the value of ρ

_{∞}has a slight influence on the phase accuracy of the proposed methods; therefore, we only determine m

_{cr}for the AECM1.

_{∞}= 1 have the forms as

**S**(∆t

_{N}) as

_{0}= 1, and $\dot{x}$

_{0}= 1; ∆t = 1 and ∆t

_{N}= 1/N are used in the AEC/DM1 and TR, respectively. The absolute errors in displacement, velocity, and acceleration of the AECM1 and TR are drawn in Figure 7, and one can see that: (a) When m > m

_{cr}= 27 is achieved from Equation (57), the absolute errors of the AECM1 trends with computer precision; (b) With the increase of m, the accuracy of TR increases for the case of m < 20, and then its accuracy begins to decline when m > 20 due to the rounding errors. Figure 8 plots the relative errors of the two methods for the damped case, in which ξ = 0.5, ω = 2π, x

_{0}= 1, and $\dot{x}$

_{0}= 0, and it follows that the accuracy of the AECM1 has no considerable variation when m > m

_{cr}= 27. Therefore, one can conclude that the selection of m

_{cr}can only consider the undamped case. The m

_{cr}corresponding to different ω∆t is given in Table 2, in which one can find that with the increase of ω∆t, the m

_{cr}becomes larger.

_{0}= 57/65, $\dot{x}$

_{0}= 2/65, and f(t) = sin(2t) are used. The absolute errors of the AECM1 and the TR are compared in Figure 9, in which one can see that: (a) The AECM1 has the same convergence rates with the second-order-accurate TR before m < 22; (b) Due to the rounding errors, the accuracy of the TR begins to decrease after m ≥ 23; (c) The accuracy of the AECM1 trends to constants with the decrease in time-step size, and the accuracy of the AECM1 with four Gauss–Legendre nodes is close to computer precision after m > m

_{cr}= 23 is achieved from Equation (57). Therefore, four Gauss–Legendre nodes are suggested for the AEC/DM1.

#### 3.2. Stability, Dissipation, and Accuracy of the Fourth-Order Scheme

#### 3.2.1. Spectral Characteristics

_{∞}.

_{∞}= 0) and AECM2 versus τ for the undamped case are shown in Figure 11 and Figure 12, respectively. It can be seen that: (a) With the increase in m, the amplitude and period errors can be simultaneously decreased; (b) From the comparison between Figure 2 and Figure 11, one can find that the accuracy, including amplitude and phase of the AEDM2, is far higher than that of the AEDM1 (ρ

_{∞}= 0); (c) One can observe by comparing Figure 3 and Figure 12 that the AECM2 has a considerable phase advantage compared with the AECM1; (d) For the same m, the phase accuracy of the non-dissipative scheme and the dissipative schemes are nearly the same, implying that m mainly affect the amplitude accuracy of the AEC/DM2.

#### 3.2.2. Rounding Errors

_{cr}; thus, only the undamped case (ξ = 0) is considered for the present scheme. By comparing Equation (58)–(61) with Equation (51)–(54), we have the relative sizes between four truncation terms and four main terms in

**S**(∆t

_{N}) by

_{0}= 1, and $\dot{x}$

_{0}= 1. The absolute errors in displacement, velocity, and acceleration of the AECM2 and the Fox–Goodwin method are drawn in Figure 16, in which ∆t = 1 and ∆t

_{N}= 1/N are used in the AEC/DM2 and Fox–Goodwin method, respectively. The well-known Fox–Goodwin method is fourth-order accurate for the undamped system, whereas it is third-order accurate for the damped case. One can see from Figure 16 that the AECM2 and the Fox–Goodwin method have the same slope before m < 13, meaning that the AECM2 is strictly fourth-order accurate. Additionally, one can find that: (a) When m > m

_{cr}= 13, the absolute errors of the AECM2 trend to constants; (b) With the increase of m, the accuracy of the Fox–Goodwin method increases when m < 13, and then its accuracy begins to decline when m ≥ 13.

_{0}= 1, and $\dot{x}$

_{0}= 0. One can see from Figure 17 that the AECM2 is fourth-order accurate, but the Fox–Goodwin method turns out to be third-order accurate due to the presence of physical damping. In addition, it follows that the accuracy of the AECM2 has no observable variation when m > m

_{cr}= 13; thus, m

_{cr}given in Equation (64) is suitable for the analysis of damped dynamic systems. The critical values of m of the AECM2 are provided in Table 3, wherein one can find that compared with the second-order AECM1, the AECM2 has a smaller m

_{cr}for the same ω∆t. As shown in Figure 18, when ω∆t < 1, the m

_{cr}of the AECM2 is about 1/2~1/4 that of the AECM1, implying that the AECM2 enjoys an advantage in efficiency when applied to dynamic systems wherein the low-frequency modes dominate.

_{0}= 57/65, $\dot{x}$

_{0}= 2/65, and f(t) = sin(2t) are adopted. The absolute errors of the AECM2 and the Fox–Goodwin method are compared in Figure 19. It can be seen that: (a) The AECM2 is fourth-order accurate for the dynamic systems including external excitation; (b) Due to the rounding errors, the accuracy of the Fox–Goodwin method begins to decrease after m > 24; (c) The accuracy of the AECM2 trend to constants with the decrease in time-step size, and together with four Gauss–Legendre nodes, the accuracy of AECM2 is close to computer precision after m > m

_{cr}= 10 is achieved from Equation (64). Then, four Gauss–Legendre nodes are employed in the AEC/DM2.

#### 3.3. Overshoot Characteristics

_{0}= 1 and $\dot{x}$

_{0}= 0 is considered for testing overshooting behavior. Figure 20 and Figure 21 draw the displacement and velocity of the AEC/DM1 and the AEC/DM2, respectively, at the first step versus ∆t/T, and numerical results validate that our methods have no overshoots both in displacement and velocity.

## 4. Numerical Experiments

#### 4.1. Linear Systems

#### 4.1.1. Stiff System

_{1}(0) = 5, y

_{2}(0) = $-$5, and the theoretical solutions are

_{1}of all methods are shown in Figure 22, in which one can find that: (a) Among these single-step methods, the higher-order accurate Fox–Goodwin method is unstable for the larger time step size ∆t = 0.1 due to intrinsic conditional stability, while other methods are convergent; (b) With the increase of m, the proposed methods’ accuracy can be noticeably improved; (c) The AECM1 (m > 25) and AECM2 (m > 15) converge to computer precision 10

^{−16}, validating that the m

_{cr}given in Section 3 is reliable; (d) With the decrease of step size, the TR and Fox–Goodwin methods can both obtain higher accuracy, but their accuracy is far lower than that of the proposed methods.

_{2}of these methods are drawn in Figure 23, and some new phenomena can be found. With physical damping, the first-order overshooting components enter into TR, meaning that they induce obvious oscillations for the larger time step size. Since the proposed methods have no overshoots both for undamped and damped systems, they can accurately simulate dynamic problems, including stiff modes.

**A**

_{num}(Δt

**H**), and ‘Recursions’ represents recursive computations of all time steps. Considering that the DOFs of this example are only two, we only discuss the effect of the value of m and the size of the time step on computational costs. From Table 3, we can find that: (a) The value of m has little effect on the computations for the proposed methods, meaning that the proposed methods’ can accuracy be enhanced without efficiency loss; (b) With the decrease of time step size, the accuracy of the TR and Fox–Goodwin methods can be slowly improved, and they need considerable computational costs.

#### 4.1.2. Cantilever Plane Truss

^{11}N/m

^{2}, ρ = 1.78 × 10

^{3}kg/m

^{3}, and A = 1.96 × 10

^{−3}m

^{2}, respectively. In this example, ∆t = 10

^{−5}is used in the proposed methods, and ∆t = 10

^{−5}, ∆t = 10

^{−6}, and ∆t = 10

^{−7}are employed in the TR and Fox–Goodwin methods. The maximum natural frequency of the cantilever plane truss is 24,378, and then the m

_{cr}for the AECM1 and the AECM2 can be determined from Section 3, which are 22 and 8, respectively.

_{cr}; (b) The accuracy improvement of the proposed methods can be achieved by increasing m, while other methods can increase accuracy by decreasing the step size; (c) Among them, the second-order accuracy TR exhibits observable numerical errors in the simulations of velocities and accelerations.

^{−6}have the same costs with those of the proposed methods, but our methods perform with higher accuracy.

#### 4.2. Nonlinear Systems

#### 4.2.1. Averaged System in Wind-Induced Oscillation

_{1}(0) = 0 and x

_{2}(0) = 1, and r = 200. To improve the stability of these proposed methods in solving dynamic systems, including geometric nonlinearity, ρ

_{∞}= 0 is utilized. The AEDM1 and AEDM2 adopt ∆t = 1/(2r) and ∆t = 1/r, respectively; ∆t = 1/(8r), ∆t = 1/(15r) and ∆t = 1/(20r) are used in the modified Euler method; ∆t = 1/(4r), ∆t = 1/(10r) and ∆t = 1/(15r) are used in the RK4 method. The relative values of energy errors of these methods are plotted in Figure 28, and Table 6 provides their computations. One can find that: (a) With the increase of the m, the accuracy of the AEDM1 and AEDM2 can be obviously enhanced without additional burdens; (b) The accuracy of the modified Euler method and the RK4 method can be improved by decreasing the size of time step; (c) When the computations of these methods are the roughly same, the AEDM2 is four orders of magnitude more accurate than the modified Euler method and the RK method.

_{cr}can be obtained from Table 2 and Table 3, and m

_{cr}= 22~26 for AEDM1 and m

_{cr}= 12 for AEDM2. From Figure 28, one can observe that the AEDM1 with m > 25 and the AEDM2 with m > 10 have no observable accuracy improvements. In a way, as applied to nonlinear systems, the proposed methods can select appropriate values of m in terms of the initial dynamic characteristic of nonlinear problems.

#### 4.2.2. Seven-Story Shear Building with Bouc–Wen Hysteresis Model

**z**represents the hysteretic displacement vector related to the displacement

**x**; $\ddot{\mathit{x}}$

_{g}is the vector of ground motion acceleration which has the form of $\ddot{\mathit{x}}$

_{g}= [

**0**

^{T}sin(t)]

^{T}. In this model, the damping matrix

**C**is defined as

_{1}and ξ

_{2}are the viscous damping ratios, which are assumed to be 3% and 5%, respectively, and ω

_{1}and ω

_{2}are the first two frequencies. The hysteretic displacement vector

**z**is formulated as

_{z}, γ

_{z,}and n

_{z}stand for the hysteretic shape parameters; h(z

_{l}) stands for the pinching function; v

_{z}(t) and η

_{z}(t) are the strength and stiffness degradation functions, respectively. The v

_{z}(t) and η

_{z}(t) can be obtained by setting

_{v}and δ

_{η}represent the strength and stiffness degradation ratios, respectively; the hysteretic energy function ε(t) has the form of

_{z,l}(t) represents the associated internal hysteretic force variables collected in the hysteretic restoring force vector (1-μ)

**Kz**(t). The pinching function h(z

_{l}) is formulated as

_{z}represents a constant that sets the pinching level as a fraction of z

_{max}; z

_{u,l}is the ultimate value of z

_{l}(t), which is obtained by

_{1}(t) controls the magnitude of the initial drop in slope, which is given by

_{z}is a constant that contributes to the rate of the initial drop in slope and ζ

_{s}is the measure of total slip. The ζ

_{2}(t) causes the pinching region to spread, and its expression is

_{ψ}represent the pinching magnitude and rate, respectively; λ is a parameter that controls the variation rate of ζ

_{2}(t) with a change of ζ

_{1}(t). The hysteretic model is controlled by the above-mentioned 13 parameters, which are {A, μ, β

_{z}, γ

_{z}, n

_{z}, δ

_{v}, δ

_{η}, ζ

_{s}, p

_{z}, q

_{z}, ψ, δ

_{ψ}, λ} wherein {μ, β

_{z}, γ

_{z}, n

_{z}} determine the shape of the hysteretic model; {δ

_{v}, δ

_{η}} control the system degradation; {ζ

_{s}, p

_{z}, q

_{z}, ψ, δ

_{ψ}, λ} control the pinching phenomenon. In this example, these parameters are assumed to be A = 1, μ = 0.02, β

_{z}= 100, γ

_{z}= 100, n

_{z}= 1.1, δ

_{v}= 0.02, δ

_{η}= 0.1, p

_{z}= 0.02, q

_{z}= 0.3, ζ

_{s}= 0.9, ψ = 0.1, δ

_{ψ}= 0.11, and λ = 0.1.

#### 4.2.3. N-Degree-of-Freedom Mass-Spring System

^{5}N/m and α = −2. Additionally, all masses are subjected to the external forces of f

_{i}(t) = m

_{i}sin(t) (i = 1, …, N). In this example, two advanced time integration methods, the LMS2 [60] and the ρ

_{∞}-Bathe method [10], are considered for comparison. The two-step LMS2 is especially effective for stiff systems, and the two-sub-step ρ

_{∞}-Bathe method has been integrated with ADINA due to its superior properties. With zero initial conditions, two cases of N = 100 and 1000 are simulated.

_{∞}-Bathe method (Δt = 0.002 s) are drawn in Figure 36 and Figure 37, and their computations are provided in Table 8 and Table 9. It can be concluded that under the same accuracy performances and compared with the LMS2 and the ρ

_{∞}-Bathe method, the AECM1 and the AECM2 perform considerable advantages in computational efficiency, especially for dynamic problems containing large degree-of-freedoms.

## 5. Conclusions

^{m}algorithm and the storage of an incremental matrix method were adopted in the proposed methods to quickly and accurately transfer the response of the linear responses, and the nonlinear responses are approximated by the Gauss–Legendre quadrature and explicit Runge–Kutta method. For linear-free vibration problems, the proposed methods can converge to computer precision, and for linear-forced vibration problems and nonlinear problems, the proposed methods enjoy considerable advantages both in accuracy and efficiency compared with the widely-used time integration methods.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Spectral radius of the AEC/DM1 versus τ for the undamped case: (

**a**) AEDM1, ρ

_{∞}= 0; (

**b**) AEDM1, ρ

_{∞}= 0.3; (

**c**) AEDM1, ρ

_{∞}= 0.7; and (

**d**) AECM1, ρ

_{∞}= 1.

**Figure 2.**Numerical damping ratio and period elongation of the AEDM1 (ρ

_{∞}= 0) versus τ for the undamped case.

**Figure 7.**Relative errors in displacement, velocity, and acceleration of the AECM1 for the case of ξ = 0 and f(t) = 0.

**Figure 8.**Relative errors in displacement, velocity, and acceleration of the AECM1 for the case of ξ = 0.5 and f(t) = 0.

**Figure 9.**Relative errors in displacement, velocity, and acceleration of the AECM1 for the case of ξ = 2/$\sqrt{5}$) and f(t) = sin(2t) (r represents number of Gauss–Legendre nodes).

**Figure 10.**Spectral radius of the AEC/DM2 versus τ for the undamped case: (

**a**) AEDM2, ρ

_{∞}= 0; (

**b**) AEDM2, ρ

_{∞}= 0.3; (

**c**) AEDM2, ρ

_{∞}= 0.7; and (

**d**) AECM2, ρ

_{∞}= 1.

**Figure 11.**Numerical damping ratio and period elongation of the AEC/DM2 (ρ

_{∞}= 0) versus τ for the undamped case.

**Figure 16.**Relative errors in displacement, velocity, and acceleration of the AECM2 for the case of ξ = 0 and f(t) = 0.

**Figure 17.**Relative errors in displacement, velocity, and acceleration of the AECM2 for the case of ξ = 0.5 and f(t) = 0.

**Figure 19.**Relative errors in displacement, velocity, and acceleration of the AECM2 for the case of ξ = 2/$\sqrt{5}$ and f(t) = sin(2t) (r represents number of Gauss–Legendre nodes).

**Figure 22.**Displacement at the first node and absolute errors versus time: (

**a**) AECM1; (

**b**) AECM2; (

**c**) TR; and (

**d**) Fox–Goodwin.

**Figure 23.**Displacement at the second node and absolute errors versus time: (

**a**) AECM1; (

**b**) AECM2; (

**c**) TR; and (

**d**) Fox–Goodwin.

**Figure 25.**Displacement of node 52 in vertical direction and absolute errors: (

**a**) AECM1; (

**b**) AECM2; (

**c**) TR; and (

**d**) Fox–Goodwin.

**Figure 26.**Velocity of node 52 in a vertical direction and absolute errors: (

**a**) AECM1; (

**b**) AECM2; (

**c**) TR; and (

**d**) Fox–Goodwin.

**Figure 27.**Acceleration of node 52 in a vertical direction and absolute errors: (

**a**) AECM1; (

**b**) AECM2; (

**c**) TR; and (

**d**) Fox–Goodwin.

**Figure 28.**Energy errors of these methods versus time: (

**a**) Euler; (

**b**) RK4; (

**c**) AEDM1; and (

**d**) AEDM2.

**Figure 30.**Numerical results of the modified Euler method at the bottom story: (

**a**) Numerical results of the bottom story in the [0, 30]s; (

**b**) Velocity of the bottom story in the [27, 29]s.

**Figure 31.**Numerical results of the RK4 method at the bottom story: (

**a**) Numerical results of the bottom story in the [0, 30]s; (

**b**) Velocity of the bottom story in the [27, 29]s.

**Figure 32.**Numerical results of the AECM1 at the bottom story: (

**a**) Numerical results of the bottom story in the [0, 30]s; (

**b**) Velocity of the bottom story in the [27, 29]s.

**Figure 33.**Numerical results of the AECM2 at the bottom story: (

**a**) Numerical results of the bottom story in the [0, 30]s; (

**b**) Velocity of the bottom story in the [27, 29]s.

**Figure 34.**Relative errors of the AECM1 and AECM2 in displacement at the bottom story: (

**a**) AECM1; and (

**b**) AECM2.

**Figure 36.**Displacement of the Nth mass (N = 100): (

**a**) Simulations in the interval [0, 30]s; (

**b**) Simulations in the interval [27.44, 27.56]s.

**Figure 37.**Displacement of the Nth mass (N = 1000): (

**a**) Simulations in the interval [0, 30]s; (

**b**) Simulations in the interval [25.44, 25.56]s.

**Table 1.**Absolute errors of the method of storing total matrix and the method of storing incremental matrix.

m = 1 | m = 10 | m = 100 | m = 1000 | |
---|---|---|---|---|

Total | 0.00267091807564768 | 5.39597800242042 × 10^{−6} | 0.105170918075648 | 0.105170918075648 |

increment | 0.00267091807564768 | 5.39597790183422 × 10^{−6} | 0 | 0 |

ω∆t | 0.01 | 0.1 | 1 | 10 | 20 | 50 | 100 | 1000 | 10,000 |

m_{cr} | 19 | 22 | 26 | 29 | 30 | 31 | 32 | 36 | 39 |

ω∆t | 0.01 | 0.1 | 1 | 10 | 20 | 50 | 100 | 1000 | 10,000 |

m_{cr} | 5 | 8 | 12 | 15 | 16 | 17 | 18 | 22 | 25 |

AECM1 | AECM2 | |||||||
---|---|---|---|---|---|---|---|---|

m = 5 | m = 10 | m = 20 | m = 25 | m = 5 | m = 10 | m = 20 | m = 25 | |

Preparation | 5.7800 × 10^{−5} | 6.2700 × 10^{−5} | 6.3500 × 10^{−5} | 6.6700 × 10^{−5} | 0.0016 | 0.0017 | 0.0018 | 0.0016 |

Recursion | 0.0177 | 0.0135 | 0.0139 | 0.0138 | 0.0144 | 0.0125 | 0.0125 | 0.0109 |

Total | 0.0178 | 0.0136 | 0.0140 | 0.0139 | 0.0160 | 0.0142 | 0.0143 | 0.0125 |

TR | Fox–Goodwin | |||||||

∆t = 0.1 | ∆t = 0.01 | ∆t = 0.001 | ∆t = 0.1 | ∆t = 0.01 | ∆t = 0.001 | |||

Preparation | - | - | - | - | - | - | ||

Recursion | 0.0100 | 0.1251 | 1.8498 | 0.0122 | 0.1063 | 1.3964 | ||

Total | 0.0100 | 0.1251 | 1.8498 | 0.0122 | 0.1063 | 1.3964 |

AECM1 | AECM2 | |||||||
---|---|---|---|---|---|---|---|---|

m = 5 | m = 10 | m = 20 | m = 25 | m = 5 | m = 10 | m = 20 | m = 25 | |

Preparation | 0.0183 | 0.0254 | 0.0353 | 0.0385 | 0.0209 | 0.0286 | 0.0383 | 0.0459 |

Recursion | 2.2373 | 2.4461 | 2.8605 | 2.8807 | 2.7853 | 2.6430 | 2.9451 | 2.9320 |

Total | 2.2556 | 2.4715 | 2.8958 | 2.9192 | 2.8062 | 2.6716 | 2.9834 | 2.9779 |

TR | Fox–Goodwin | |||||||

∆t = 10^{−5} | ∆t = 10^{−6} | ∆t = 10^{−7} | ∆t = 10^{−5} | ∆t = 10^{−6} | ∆t = 10^{−7} | |||

Preparation | - | - | - | - | - | - | ||

Recursion | 0.1629 | 1.8562 | 26.8233 | 0.1306 | 1.8313 | 21.9914 | ||

Total | 0.1629 | 1.8562 | 26.8233 | 0.1306 | 1.8313 | 21.9914 |

AEDM1 | AEDM2 | |||||
---|---|---|---|---|---|---|

m = 20 | m = 25 | m = 30 | m = 5 | m = 10 | m = 15 | |

Preparation | 0.000169 | 0.000140 | 0.000396 | 0.002044 | 0.001902 | 0.002686 |

Recursion | 4.006249 | 4.395022 | 4.950563 | 4.867872 | 5.423115 | 6.616388 |

Total | 4.006418 | 4.395162 | 4.950959 | 4.869916 | 5.425017 | 6.619074 |

Euler | RK4 | |||||

∆t = 1/(8r) | ∆t = 1/(15 r) | ∆t = 1/(20 r) | ∆t = 1/(4 r) | ∆t = 1/(10 r) | ∆t = 1/(15 r) | |

Preparation | - | - | - | - | - | - |

Recursion | 4.054805 | 6.3309 | 10.699817 | 7.299833 | 16.232617 | 26.404818 |

Total | 4.054805 | 6.3309 | 10.699817 | 7.299833 | 16.232617 | 26.404818 |

AECM1 | AECM2 | |||||
---|---|---|---|---|---|---|

m = 5 | m = 10 | m = 20 | m = 5 | m = 10 | m = 20 | |

Preparation | 7.2980 × 10^{−4} | 5.4800 × 10^{−4} | 7.3120 × 10^{−4} | 0.0027 | 0.0029 | 0.0029 |

Recursion | 25.7393 | 26.2247 | 27.3551 | 23.0725 | 23.5911 | 25.9989 |

Total | 25.7400 | 26.2252 | 27.3558 | 23.0752 | 23.5940 | 26.0018 |

Euler | RK4 | |||||

∆t = 0.001 | ∆t = 0.0002 | ∆t = 0.0001 | ∆t = 0.002 | ∆t = 0.0004 | ∆t = 0.0002 | |

Preparation | - | - | - | - | - | - |

Recursion | 16.3542 | 75.2145 | 155.6917 | 14.6172 | 58.4983 | 139.2688 |

Total | 16.3542 | 75.2145 | 155.6917 | 14.6172 | 58.4983 | 139.2688 |

AECM1 | AECM2 | LMS2 | ρ_{∞}-Bathe | |
---|---|---|---|---|

Preparation | 0.0197 | 0.0184 | - | - |

Recursion | 3.7715 | 2.4026 | 19.8547 | 19.5784 |

Total | 3.7912 | 2.4210 | 19.8547 | 19.5784 |

AECM1 | AECM2 | LMS2 | ρ_{∞}-Bathe | |
---|---|---|---|---|

Preparation | 13.6961 | 22.8801 | - | - |

Recursion | 339.4413 | 248.7944 | 820.9083 | 686.0195 |

Total | 353.1374 | 271.6745 | 820.9083 | 686.0195 |

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**MDPI and ACS Style**

Ji, Y.; Xing, Y.
Highly Accurate and Efficient Time Integration Methods with Unconditional Stability and Flexible Numerical Dissipation. *Mathematics* **2023**, *11*, 593.
https://doi.org/10.3390/math11030593

**AMA Style**

Ji Y, Xing Y.
Highly Accurate and Efficient Time Integration Methods with Unconditional Stability and Flexible Numerical Dissipation. *Mathematics*. 2023; 11(3):593.
https://doi.org/10.3390/math11030593

**Chicago/Turabian Style**

Ji, Yi, and Yufeng Xing.
2023. "Highly Accurate and Efficient Time Integration Methods with Unconditional Stability and Flexible Numerical Dissipation" *Mathematics* 11, no. 3: 593.
https://doi.org/10.3390/math11030593