Comparison of Semi-Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations
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Comparison of Semi‑Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations Xiaofeng Cai1 · Wei Guo2 · Jing‑Mei Qiu1 Received: 31 March 2020 / Revised: 23 June 2020 / Accepted: 8 July 2020 © Shanghai University 2020
Abstract Transport problems arise across diverse fields of science and engineering. Semi-Lagrangian (SL) discontinuous Galerkin (DG) methods are a class of high-order deterministic transport solvers that enjoy advantages of both the SL approach and the DG spatial discretization. In this paper, we review existing SLDG methods to date and compare numerically their performance. In particular, we make a comparison between the splitting and nonsplitting SLDG methods for multi-dimensional transport simulations. Through extensive numerical results, we offer a practical guide for choosing optimal SLDG solvers for linear and nonlinear transport simulations. Keywords Semi-Lagrangian (SL) · Discontinuous Galerkin (DG) · Transport simulations · Splitting · Non-splitting · Comparison Mathematics Subject Classification 65M60 · 65M25
1 Introduction Semi-Lagrangian (SL) discontinuous Galerkin (DG) methods are a class of high-order transport solvers that enjoy the computational advantages of both the SL approach and the DG spatial discretization. In this paper, we conduct a systematic comparison for several existing SLDG methods in the literature by considering aspects including the accuracy, W. Guo: Research is supported by NSF grant NSF-DMS-1830838. J.-M. Qiu: Research is supported by NSF grant NSF-DMS-1522777 and NSF-DMS-1818924, Air Force Office of Scientific Computing FA9550-18-1-0257. * Wei Guo [email protected] Xiaofeng Cai [email protected] Jing‑Mei Qiu [email protected] 1
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA
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Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 70409, USA
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Vol.:(0123456789)
Communications on Applied Mathematics and Computation
CPU efficiency, conservation properties, implementation difficulties, with the aim to provide a brief survey of the recent development along this line of research for simulating linear and nonlinear transport problems. The DG methods belong to the family of finite element methods, which employ piecewise polynomials as approximation and test function spaces. Such methods have undergone rapid development for simulating partial differential equations (PDEs) over the last few decades. For the time-dependent transport simulations, the DG methods are often coupled with the method-of-lines Eulerian framework, using appropriate time integrators for time evolution, e.g., the well-known TVD Runge-Kutta (RK) methods. It is well known that the DG method, when coupled with an explicit time integrator, suffers a very stringent CFL time step restriction for stability, which may be much smaller than that needed to resolve interesting timescales in physics. Implicit methods can be used to avoid the CFL time step restriction, yet the additional computational cost is req
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