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A Third Order Adaptive ADER Scheme for One Dimensional Conservation Laws

Part of: Hyperbolic equations and systems Equations of mathematical physics and other areas of application Partial differential equations, initial value and time-dependent initial-boundary value problems Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  06 July 2017

Yaguang Gu  and
Guanghui Hu
Yaguang Gu*
Affiliation:
Department of Mathematics, University of Macau, Macao SAR, China
Guanghui Hu*
Affiliation:
Department of Mathematics, University of Macau, Macao SAR, China UM Zhuhai Research Institute, Zhuhai, Guangdong, China
*
*Corresponding author. Email addresses:umacgyg@gmail.com (Y. G. Gu), garyhu@umac.mo (G. H. Hu)
*Corresponding author. Email addresses:umacgyg@gmail.com (Y. G. Gu), garyhu@umac.mo (G. H. Hu)
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Abstract

We introduce a third order adaptive mesh method to arbitrary high order Godunov approach. Our adaptive mesh method consists of two parts, i.e., mesh-redistribution algorithm and solution algorithm. The mesh-redistribution algorithm is derived based on variational approach, while a new solution algorithm is developed to preserve high order numerical accuracy well. The feature of proposed Adaptive ADER scheme includes that 1). all simulations in this paper are stable for large CFL number, 2). third order convergence of the numerical solutions is successfully observed with adaptive mesh method, and 3). high resolution and non-oscillatory numerical solutions are obtained successfully when there are shocks in the solution. A variety of numerical examples show the feature well.

Keywords

ADER schemeadaptive mesh methodfinite volume methodWENO reconstructionhyperbolic equations

MSC classification

Secondary: 35L65: Conservation laws 35L67: Shocks and singularities 58J45: Hyperbolic equations 35Q31: Euler equations 65M08: Finite volume methods
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 3 , September 2017 , pp. 829 - 851
Copyright
Copyright © Global-Science Press 2017 

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