scholarly journals On the Convergence of a Shock Capturing Discontinuous Galerkin Method for Nonlinear Hyperbolic Systems of Conservation Laws

2016 ◽  
Vol 54 (2) ◽  
pp. 874-898 ◽  
Author(s):  
Mohammad Zakerzadeh ◽  
Georg May
Keyword(s):  
Conservation Laws ◽  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Hyperbolic Systems ◽  
Shock Capturing ◽  
Systems Of Conservation Laws ◽  
Nonlinear Hyperbolic Systems

Efficient Preconditioners for a Shock Capturing Space-Time Discontinuous Galerkin Method for Systems of Conservation Laws

2015 ◽  
Vol 17 (5) ◽  
pp. 1320-1359 ◽  
Author(s):  
Andreas Hiltebrand ◽  
Siddhartha Mishra
Keyword(s):  
Conservation Laws ◽  
Discontinuous Galerkin ◽  
Hyperbolic Systems ◽  
Approximate Solutions ◽  
Space Time ◽  
Shock Capturing ◽  
Algebraic Equations ◽  
Time Step ◽  
Newton Step ◽  
Systems Of Conservation Laws

AbstractAn entropy stable fully discrete shock capturing space-time Discontinuous Galerkin (DG) method was proposed in a recent paper to approximate hyperbolic systems of conservation laws. This numerical scheme involves the solution of a very large nonlinear system of algebraic equations, by a Newton-Krylov method, at every time step. In this paper, we design efficient preconditioners for the large, non-symmetric linear system, that needs to be solved at every Newton step. Two sets of preconditioners, one of the block Jacobi and another of the block Gauss-Seidel type are designed. Fourier analysis of the preconditioners reveals their robustness and a large number of numerical experiments are presented to illustrate the gain in efficiency that results from preconditioning. The resulting method is employed to compute approximate solutions of the compressible Euler equations, even for very high CFL numbers.

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