### Electronic Research Archive

2021, 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

• 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.

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

### Related Papers:

• 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.

 [1] S. Boscarino, S.-Y. Cho, G. Russo and S.-B. Yun, High order conservative semi-Lagrangian scheme for the BGK model of the Boltzmann equation, submitted (2019), arXiv: 1905.03660. [2] X. Cai, S. Boscarino and J.-M. Qiu, High order semi-Lagrangian discontinuous galerkin method coupled with Runge-Kutta exponential integrators for nonlinear Vlasov dynamics, submitted (2019), arXiv: 1911.12229. [3] A high order conservative semi-Lagrangian discontinuous Galerkin method for two-dimensional transport simulations. Journal of Scientific Computing (2017) 73: 514-542. [4] A conservative semi-Lagrangian HWENO method for the Vlasov equation. Journal of Computational Physics (2016) 323: 95-114. [5] Semi-Lagrangian Runge-Kutta exponential integrators for convection dominated problems. Journal of Scientific Computing (2009) 41: 139-164. [6] High order semi-Lagrangian methods for the incompressible Navier–Stokes equations. Journal of Scientific Computing (2016) 66: 91-115. [7] Commutator-free Lie group methods. Future Generation Computer Systems (2003) 19: 341-352. [8] High order semi-Lagrangian particles for transport equations: Numerical analysis and implementation issues. ESIAM: Mathematical Modelling and Numerical Analysis (2014) 48: 1029-1060. [9] Conservative semi-Lagrangian schemes for Vlasov equations. Journal of Computational Physics (2010) 229: 1927-1953. [10] A forward semi-Lagrangian method for the numerical solution of the Vlasov equation. Computer Physics Communications (2009) 180: 1730-1745. [11] Implicit scheme for hyperbolic conservation laws using nonoscillatory reconstruction in space and time. SIAM Journal on Scientific Computing (2007) 29: 2607-2620. [12] A. Efremov, E. Karepova and V. Shaydurov, A conservative semi-Lagrangian method for the advection problem, Numerical Analysis and its Applications, Lecture Notes in Comput. Sci., Springer, Cham, 10187 (2017), 325–333. doi: 10.1007/978-3-319-57099-0 [13] A semi-Lagrangian spectral method for the Vlasov-Poisson system based on Fourier, Legendre and Hermite polynomials. Communications on Applied Mathematics and Computation (2019) 1: 333-360. [14] High order time discretization for backward semi-Lagrangian methods. Journal of Computational and Applied Mathematics (2016) 303: 171-188. [15] A conservative semi-Lagrangian discontinuous Galerkin scheme on the cubed-sphere. Monthly Weather Review (2014) 142: 457-475. [16] A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws. Journal of Computational Physics (2016) 322: 559-585. [17] R. J. LeVeque, Numerical Methods for Conservation Laws, Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1992. doi: 10.1007/978-3-0348-8629-1 [18] High order mass conservative semi-Lagrangian methods for transport problems. Handbook of Numerical Methods for Hyperbolic Problems (2016) 17: 353-382. [19] A conservative high order semi-Lagrangian WENO method for the Vlasov equation. Journal of Computational Physics (2010) 229: 1130-1149. [20] A high order multi-dimensional characteristic tracing strategy for the Vlasov-Poisson System. Journal of Scientific Computing (2017) 71: 414-434. [21] Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow. Journal of Computational Physics (2011) 230: 863-889. [22] Convergence of Godunov-type schemes for scalar conservation laws under large time steps. SIAM Journal on Numerical Analysis (2008) 46: 2211-2237. [23] A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations. Journal of Computational Physics (2011) 230: 6203-6232. [24] High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Review (2009) 51: 82-126. [25] A truly forward semi-Lagrangian WENO scheme for the Vlasov-Poisson system. Journal of Computational Physics (2019) 392: 619-665. [26] The semi-Lagrangian method for the numerical resolution of the Vlasov equation. Journal of Computational Physics (1999) 149: 201-220. [27] Semi-Lagrangian integration schemes for atmospheric models: A review. Monthly Weather Review (1991) 119: 2206-2223. [28] A semi-implicit, semi-Lagrangian, $p$-adaptive discontinuous Galerkin method for the shallow water equations. Journal of Computational Physics (2013) 232: 46-67. [29] High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation. Journal of Computational Physics (2014) 273: 618-639. [30] High order multi-dimensional characteristics tracing for the incompressible Euler equation and the guiding-center Vlasov equation. Journal of Scientific Computing (2018) 77: 263-282. [31] Conservative multi-dimensional semi-Lagrangian finite difference scheme: Stability and applications to the kinetic and fluid simulations. Journal of Scientific Computing (2019) 79: 1241-1270. [32] A semi-Lagrangian high-order method for Navier-Stokes equations. Journal of Computational Physics (2001) 172: 658-684. [33] Conservative semi-Lagrangian CIP technique for the shallow water equations. Computational Mechanics (2010) 46: 125-134.
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沈阳化工大学材料科学与工程学院 沈阳 110142

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