# High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Modified Engquist–Osher-Type Interface Numerical Flux

**(A1)**$f,\phantom{\rule{4pt}{0ex}}g\in Lip\left(\right[0,1\left]\right)$, $f\left(u\right)=g\left(u\right)=0$ for $u=0,1$, f has a single maximum at ${\theta}_{f}\in [0,1]$ and g has a single maximum at ${\theta}_{g}\in [0,1]$. The function f (or g) is strictly increasing on $(0,{\theta}_{f})$ (or $(0,{\theta}_{g})$) and strictly decreasing on $({\theta}_{f},1)$ (or $({\theta}_{g},1)$).

**(A2)**both f and g are genuinely nonlinear in the sense that f and g are not linear on any nondegenerate interval.

**(A3)**there exists at most one point $\widehat{u}\in (0,1)$ such that $f\left(\widehat{u}\right)=g\left(\widehat{u}\right)$.

**Lemma**

**1.**

- (1)
- $\tilde{h}(x,y)$ is nondecreasing function in its first variable, nonincreasing function in its second variable, and Lipschitz continuous on both two arguments.
- (2)
- $\tilde{h}(x,y)$ is not consistent, but $\tilde{h}(0,0)=0$, $\tilde{h}(1,1)=0$, and $\tilde{h}(A,B)=g\left(A\right)=f\left(B\right)$.

**Proof.**

**MEO**, where the mesh ratio $\lambda =\Delta t/\Delta x$ and the Lipschitz constant $\mathcal{M}$ is defined in (9).

## 3. High-Order Accurate Schemes for Conservation Laws (1)

#### 3.1. High-Order WENO Scheme Based on the Modified Interface Engquist–Osher-Type Flux

**MEO-WENO5**.

#### 3.2. High-Order WENO Schemes Based on the Reconstructions of the Unknown Functions

**DFLU-WENO5**.

**DFLU-WENO5B**.

## 4. Numerical Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3**

**.**Considering the two-phase incompressible flow in a porous medium with a rock type changing at $x=0$. The flux functions $g\left(u\right)$ and $f\left(u\right)$ have the following form

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The graphs of functions $f\left(x\right)$ and $g\left(x\right)$ and the $(A,B)$-connection.

**Figure 3.**Example 1: Numerical solutions computed by the MEO-1 and DFLU-1 with the different meshes. (

**a**): $\Delta x=1/25$; (

**b**): $\Delta x=1/50$.

**Figure 4.**Example 1 computed by the MEO-WENO5 and DFLU-WENO5 with $\Delta x=1/25$. (

**a**): the numerical and exact solutions; (

**b**): the numerical and exact solutions zoomed in.

**Figure 5.**Example 1 computed by the MEO-WENO5 and DFLU-WENO5 with $\Delta x=1/50$. (

**a**): the numerical and exact solutions; (

**b**): the numerical and exact solutions zoomed in.

**Figure 6.**Example 1 computed by the MEO-WENO5 and DFLU-WENO5B with different meshes. (

**a**): $\Delta x=1/25$; (

**b**): $\Delta x=1/50$.

**Figure 7.**Example 2: Numerical solutions computed by the MEO-1 and DFLU-1 with the different meshes. (

**a**): $\Delta x=1/50$; (

**b**): $\Delta x=1/100$.

**Figure 8.**Example 2 computed by the MEO-WENO5, DFLU-WENO5, and DFLU-WENO5B with $\Delta x=1/25$. (

**a**): the numerical and exact solutions; (

**b**): the numerical and exact solutions zoomed in.

**Figure 9.**Example 2 computed by the MEO-WENO5 and DFLU-WENO5 with $\Delta x=1/50$. (

**a**): the numerical and exact solutions; (

**b**): the numerical and exact solutions zoomed in.

**Figure 10.**Example 2 computed by the MEO-WENO5 and DFLU-WENO5B with $\Delta x=1/50$. (

**a**): the numerical and exact solutions; (

**b**): the numerical and exact solutions zoomed in.

**Figure 11.**Example 3: Numerical solutions computed by the MEO-1 and DFLU-1 with the different meshes. (

**a**): $\Delta x=1/50$; (

**b**): $\Delta x=1/100$.

**Figure 12.**Example 3 computed by the MEO-WENO5 and DFLU-WENO5 with $\Delta x=1/25$. (

**a**): the numerical and exact solutions; (

**b**): the numerical and exact solutions zoomed in.

**Figure 13.**Example 3 computed by the MEO-WENO5 and DFLU-WENO5B with $\Delta x=1/25$. (

**a**): the numerical and exact solutions; (

**b**): the numerical and exact solutions zoomed in.

$\mathbf{\Delta}\mathit{x}$ | MEO-1 | DFLU-1 | MEO-WENO5 | DFLU-WENO5 | DFLU-WENO5B | |||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{L}}^{1}$-Error | Order | ${\mathit{L}}^{1}$-Error | Order | ${\mathit{L}}^{1}$-Error | Order | ${\mathit{L}}^{1}$-error | Order | ${\mathit{L}}^{1}$-Error | Order | |

1/25 | 1.88 × 10${}^{-2}$ | 1.80 × 10${}^{-2}$ | 8.36 × 10${}^{-3}$ | 8.70 × 10${}^{-3}$ | 8.28 × 10${}^{-3}$ | |||||

1/50 | 1.14 × 10${}^{-2}$ | 0.7230 | 1.11 × 10${}^{-2}$ | 0.7025 | 4.57 × 10${}^{-3}$ | 0.8727 | 4.57 × 10${}^{-3}$ | 0.9278 | 4.57 × 10${}^{-3}$ | 0.8567 |

1/100 | 6.99 × 10${}^{-3}$ | 0.7049 | 6.88 × 10${}^{-3}$ | 0.6846 | 2.76 × 10${}^{-3}$ | 0.7270 | 2.63 × 10${}^{-3}$ | 0.7983 | 2.62 × 10${}^{-3}$ | 0.8023 |

1/200 | 4.48 × 10${}^{-3}$ | 0.6431 | 4.46 × 10${}^{-3}$ | 0.6274 | 1.93 × 10${}^{-3}$ | 0.5144 | 1.96 × 10${}^{-3}$ | 0.4260 | 1.89 × 10${}^{-3}$ | 0.4670 |

1/400 | 2.29 × 10${}^{-3}$ | 0.9681 | 2.25 × 10${}^{-3}$ | 0.9840 | 6.13 × 10${}^{-4}$ | 1.6555 | 6.11 × 10${}^{-4}$ | 1.6794 | 6.11 × 10${}^{-4}$ | 1.6329 |

$\mathbf{\Delta}\mathit{x}$ | MEO-1 | DFLU-1 | MEO-WENO5 | DFLU-WENO5 | DFLU-WENO5B | |||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{L}}_{1}$-Error | Order | ${\mathit{L}}_{1}$-Error | Order | ${\mathit{L}}_{1}$-Error | Order | ${\mathit{L}}_{1}$-Error | Order | ${\mathit{L}}_{1}$-Error | Order | |

1/25 | 6.52 × 10${}^{-2}$ | 6.68 × 10${}^{-2}$ | 2.69 × 10${}^{-2}$ | 2.49 × 10${}^{-2}$ | 2.18 × 10${}^{-2}$ | |||||

1/50 | 4.55 × 10${}^{-2}$ | 0.5203 | 4.66 × 10${}^{-2}$ | 0.5222 | 1.82 × 10${}^{-2}$ | 0.5617 | 1.73 × 10${}^{-2}$ | 0.5092 | 1.59 × 10${}^{-2}$ | 0.4505 |

1/100 | 3.17 × 10${}^{-2}$ | 0.5225 | 3.23 × 10${}^{-2}$ | 0.5253 | 1.36 × 10${}^{-2}$ | 0.4280 | 1.31 × 10${}^{-2}$ | 0.3933 | 1.24 × 10${}^{-2}$ | 0.3626 |

1/200 | 1.91 × 10${}^{-2}$ | 0.7295 | 1.95 × 10${}^{-2}$ | 0.7269 | 7.02 × 10${}^{-3}$ | 0.9496 | 6.84 × 10${}^{-3}$ | 0.9427 | 6.45 × 10${}^{-3}$ | 0.9428 |

1/400 | 1.13 × 10${}^{-2}$ | 0.7586 | 1.16 × 10${}^{-2}$ | 0.7562 | 3.58 × 10${}^{-3}$ | 0.9733 | 3.48 × 10${}^{-3}$ | 0.9742 | 3.29 × 10${}^{-3}$ | 0.9743 |

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**MDPI and ACS Style**

Xiang, T.; Wang, G.; Zhang, S.
High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space. *Mathematics* **2021**, *9*, 1079.
https://doi.org/10.3390/math9101079

**AMA Style**

Xiang T, Wang G, Zhang S.
High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space. *Mathematics*. 2021; 9(10):1079.
https://doi.org/10.3390/math9101079

**Chicago/Turabian Style**

Xiang, Tingting, Guodong Wang, and Suping Zhang.
2021. "High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space" *Mathematics* 9, no. 10: 1079.
https://doi.org/10.3390/math9101079