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A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions

In part I of these two papers we introduced for inviscid flow in one space dimension a discontinuous Galerkin scheme of arbitrary order of accuracy in space and time. In the second part we extend the scheme to the compressible Navier-Stokes equations in multi dimensions. It is based on a space-time...

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Published in:Journal of scientific computing 2008-03, Vol.34 (3), p.260-286
Main Authors: Gassner, G., Lörcher, F., Munz, C.-D.
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description In part I of these two papers we introduced for inviscid flow in one space dimension a discontinuous Galerkin scheme of arbitrary order of accuracy in space and time. In the second part we extend the scheme to the compressible Navier-Stokes equations in multi dimensions. It is based on a space-time Taylor expansion at the old time level in which all time or mixed space-time derivatives are replaced by space derivatives using the Cauchy-Kovalevskaya procedure. The surface and volume integrals in the variational formulation are approximated by Gaussian quadrature with the values of the space-time approximate solution. The numerical fluxes at grid cell interfaces are based on the approximate solution of generalized Riemann problems for both, the inviscid and viscous part. The presented scheme has to satisfy a stability restriction similar to all other explicit DG schemes which becomes more restrictive for higher orders. The loss of efficiency, especially in the case of strongly varying sizes of grid cells is circumvented by use of different time steps in different grid cells. The presented time accurate numerical simulations run with local time steps adopted to the local stability restriction in each grid cell. In numerical simulations for the two-dimensional compressible Navier-Stokes equations we show the efficiency and the optimal order of convergence being p +1, when a polynomial approximation of degree p is used.
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subjects Algorithms
Approximation
Compressibility
Computational Mathematics and Numerical Analysis
Flow equations
Fluid flow
Galerkin method
Inviscid flow
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Navier-Stokes equations
Polynomials
Quadratures
Spacetime
Stability
Taylor series
Theoretical
Viscous flow
title A Discontinuous Galerkin Scheme based on a Space-Time Expansion II. Viscous Flow Equations in Multi Dimensions
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