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Positivity-Preserving Runge-Kutta Discontinuous Galerkin Method on Adaptive Cartesian Grid for Strong Moving Shock

Part of: Shock waves and blast waves Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  15 February 2016

Jianming Liu ,
Jianxian Qiu ,
Mikhail Goman ,
Xinkai Li  and
Meilin Liu
Jianming Liu
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China Faculty of Technology, De Montfort University, Leicester LE1 9BH, England
Jianxian Qiu*
Affiliation:
School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computation, Xiamen University, Xiamen 361005, China
Mikhail Goman
Affiliation:
Faculty of Technology, De Montfort University, Leicester LE1 9BH, England
Xinkai Li
Affiliation:
Faculty of Technology, De Montfort University, Leicester LE1 9BH, England
Meilin Liu
Affiliation:
Shanghai Institute of Satellite Engineering, Shanghai 200240, China
*
*Corresponding author. Email address: [email protected] (J.-X. Qiu)
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Abstract

In order to suppress the failure of preserving positivity of density or pressure, a positivity-preserving limiter technique coupled with h-adaptive Runge-Kutta discontinuous Galerkin (RKDG) method is developed in this paper. Such a method is implemented to simulate flows with the large Mach number, strong shock/obstacle interactions and shock diffractions. The Cartesian grid with ghost cell immersed boundary method for arbitrarily complex geometries is also presented. This approach directly uses the cell solution polynomial of DG finite element space as the interpolation formula. The method is validated by the well documented test examples involving unsteady compressible flows through complex bodies over a large Mach numbers. The numerical results demonstrate the robustness and the versatility of the proposed approach.

Keywords

Discontinuous Galerkin methodadaptive Cartesian gridpositivity-preservingimmersed boundary methodcomplex geometry

MSC classification

Secondary: 65M50: Mesh generation and refinement 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76L05: Shock waves and blast waves
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 1 , February 2016 , pp. 87 - 110
Copyright
Copyright © Global-Science Press 2016 

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