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A Study on cfl conditions for the DG Solution of Conservation Laws on Adaptive Moving Meshes

Numerical Mathematics(Theory,Methods and Applications) 2023年第1期16卷 111-139页

作者： Min Zhang Weizhang Huang Jianxian Qiu School of Mathematical Sciences Peking UniversityBeijing 100871China Department of Mathematics University of KansasLawrenceKansas 66045USA School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical ModelingHigh-Performance Scientific ComputingXiamen UniversityXiamenFujian 361005China

The selection of time step plays a crucial role in improving stability and efficiency in the Discontinuous Galerkin(DG)solution of hyperbolic conservation laws on adaptive moving meshes that typically employs explicit stepping.A commonly used selection of time step is a direct extension based on Courant-Friedrichs-Levy(cfl)conditions established for fixed and uniform *** this work,we provide a mathematical justification for those time step selection strategies used in practical adaptive DG computations.A stability analysis is presented for a moving mesh DG method for linear scalar conservation *** on the analysis,a new selection strategy of the time step is proposed,which takes into consideration the coupling of theα-function(that is related to the eigenvalues of the Jacobian matrix of the flux and the mesh movement velocity)and the heights of the mesh *** analysis also suggests several stable combinations of the choices of theα-function in the numerical scheme and in the time step *** results obtained with a moving mesh DG method for Burgers’and Euler equations are *** comparison purpose,numerical results obtained with an error-based time step-size selection strategy are also given。

关键词： Discontinuous Galerkin method adaptive mesh moving mesh cfl condition stability

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A new fourth order finite difference scheme for the heat equation

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Communications in Nonlinear Science and Numerical Simulation 2000年第4期5卷 151-157页

作者： Yuehong QIAN, Hudong CHEN, Raoyang ZHANG and Shiyi CHEN (Exa Corporation, 450 Bedford Street, Lexington, MA 02420, USA e-mail.. qean＠*** Deportment of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21280, USA) (Received December 1, Exa Corporation Lexington MA 02420 450 Bedford Street United States Department of Mechanical Engineering Johns Hopkins University Baltimore MD 21280 United States

We propose and investigate a new simple fourth order finite difference scheme for the heat equation. Numerical simulations do confirm theoretical analysis of accuracy and stability condition.

关键词： heat equation cfl condition stability high-order scheme

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A Remark on the Courant-Friedrichs-Lewy condition in Finite Difference Approach to PDE’s

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Advances in Applied Mathematics and Mechanics 2014年第5期6卷 693-698页

作者： Kosuke Abe Nobuyuki Higashimori Masayoshi Kubo Hiroshi Fujiwara Yuusuke Iso Nihon University TokyoJapan Hitotsubashi University KunitachiJapan Graduate School of Informatics Kyoto UniversityKyotoJapan

The Courant-Friedrichs-Lewy condition(The cfl condition)is appeared in the analysis of the finite difference method applied to linear hyperbolic partial differential *** give a remark on the cfl condition from a view point of stability,and we give some numerical experiments which show instability of numerical solutions even under the cfl *** give a mathematical model for rounding errors in order to explain the instability。

关键词： Numerical analysis finite difference scheme stability cfl condition

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Efficient Finite Difference Methods for the Numerical Analysis of One-Dimensional Heat Equation

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Journal of Applied Mathematics and Physics 2023年第10期11卷 3099-3123页

作者： Md. Shahadat Hossain Mojumder Md. Nazmul Haque Md. Joni Alam Department of Mathematics Comilla University Kotbari Bangladesh Department of Mathematical Sciences Kent State University Kent OH USA

In this paper, we investigate and analyze one-dimensional heat equation with appropriate initial and boundary condition using finite difference method. Finite difference method is a well-known numerical technique for obtaining the approximate solutions of an initial boundary value problem. We develop Forward Time Centered Space (FTCS) and Crank-Nicolson (CN) finite difference schemes for one-dimensional heat equation using the Taylor series. Later, we use these schemes to solve our governing equation. The stability criterion is discussed, and the stability conditions for both schemes are verified. We exhibit the results and then compare the results between the exact and approximate solutions. Finally, we estimate error between the exact and approximate solutions for a specific numerical problem to present the convergence of the numerical schemes, and demonstrate the resulting error in graphical representation.

关键词： Explicit Scheme Implicit Scheme C-N Scheme cfl condition

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