Adaptive Moving Mesh Central-Upwind Schemes for Hyperbolic System of PDEs: Applications to Compressible Euler Equations
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Adaptive Moving Mesh Central‑Upwind Schemes for Hyperbolic System of PDEs: Applications to Compressible Euler Equations and Granular Hydrodynamics Alexander Kurganov1 · Zhuolin Qu2 · Olga S. Rozanova3 · Tong Wu2 Received: 10 July 2019 / Revised: 26 May 2020 / Accepted: 1 June 2020 © Shanghai University 2020
Abstract We introduce adaptive moving mesh central-upwind schemes for one- and two-dimensional hyperbolic systems of conservation and balance laws. The proposed methods consist of three steps. First, the solution is evolved by solving the studied system by the second-order semi-discrete central-upwind scheme on either the one-dimensional nonuniform grid or the two-dimensional structured quadrilateral mesh. When the evolution step is complete, the grid points are redistributed according to the moving mesh differential equation. Finally, the evolved solution is projected onto the new mesh in a conservative manner. The resulting adaptive moving mesh methods are applied to the one- and two-dimensional Euler equations of gas dynamics and granular hydrodynamics systems. Our numerical results demonstrate that in both cases, the adaptive moving mesh central-upwind schemes outperform their uniform mesh counterparts. Keywords Adaptive moving mesh methods · Finite-volume methods · Central-upwind schemes · Moving mesh differential equations · Euler equations of gas dynamics · Granular hydrodynamics · Singular solutions Mathematics Subject Classification 65M50 · 65M08 · 76M12 · 35L65 · 35L67
* Alexander Kurganov [email protected] Zhuolin Qu [email protected] Olga S. Rozanova [email protected] Tong Wu [email protected] 1
Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, Guangdong Province, China
2
Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA
3
Mathematics and Mechanics Faculty, Moscow State University, Moscow 119991, Russia
13
Vol.:(0123456789)
Communications on Applied Mathematics and Computation
1 Introduction We consider the hyperbolic system of conservation/balance laws:
Ut + ∇ ⋅ F(U) = S(U),
(1.1)
where U is a vector of the conserved quantities, F(U) are the flux functions, and S(U) are the source terms. The development of accurate, efficient and robust numerical methods for the system (1.1) is an important and challenging problem. A major difficulty is related to the fact that the system (1.1) admits nonsmooth solutions. Moreover, it is well-known that even smooth solutions may develop nonsmooth waves including shocks, rarefaction waves, and contact discontinuities. There is a wide variety of shock capturing methods designed to accurately capture this type of solutions; see, e.g., the monographs [3, 7, 15, 25, 31, 43] and references therein. Numerical methods for (1.1) are typically designed using fixed grids. This limits the efficiency of the methods since finer grids and special nonlinear techniques (such as nonlinear limiters) are required in
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