A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations

Guoliang Zhang; Shaoqin Zheng; Tao Xiong; Guoliang Zhang; Shaoqin Zheng; Tao Xiong

doi:10.3934/era.2020093

Electronic Research Archive

2021, Volume 29, Issue 1: 1819-1839. doi: 10.3934/era.2020093

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A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations

Guoliang Zhang ^{1
,},
Shaoqin Zheng ^{1
,},
Tao Xiong ^{2
,
,}

1.
School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, China
2.
School of Mathematical Sciences, Xiamen University, Fujian Provincial Key Laboratory of Mathematical Modeling, and High-Performance Scientific Computing, Xiamen, Fujian 361005, China

Received: 01 March 2020 Revised: 01 July 2020 Published: 16 September 2020
65M06, 65M25

In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [21], with a high order RKEI method [7], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.
- Semi-Lagrangian,
- finite difference WENO scheme,
- scalar nonlinear hyperbolic equations,
- Runge-Kutta exponential integrator,
- conservation
Citation: Guoliang Zhang, Shaoqin Zheng, Tao Xiong. A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations[J]. Electronic Research Archive, 2021, 29(1): 1819-1839. doi: 10.3934/era.2020093

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Abstract

In this paper, we propose a conservative semi-Lagrangian finite difference (SLFD) weighted essentially non-oscillatory (WENO) scheme, based on Runge-Kutta exponential integrator (RKEI) method, to solve one-dimensional scalar nonlinear hyperbolic equations. Conservative semi-Lagrangian schemes, under the finite difference framework, usually are designed only for linear or quasilinear conservative hyperbolic equations. Here we combine a conservative SLFD scheme developed in [21], with a high order RKEI method [7], to design conservative SLFD schemes, which can be applied to nonlinear hyperbolic equations. Our new approach will enjoy several good properties as the scheme for the linear or quasilinear case, such as, conservation, high order and large time steps. The new ingredient is that it can be applied to nonlinear hyperbolic equations, e.g., the Burgers' equation. Numerical tests will be performed to illustrate the effectiveness of our proposed schemes.

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