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A Compact High Order space-time method for Conservation Laws

Communications in Computational Physics 2011年第2期9卷 441-480页

作者： Shuangzhang Tu Gordon W.Skelton Qing Pang Department of Computer Engineering Jackson State UniversityJacksonMS 39217USA

This paper presents a novel high-order space-time method for hyperbolic conservation *** important concepts,the staggered space-time mesh of the space-time conservation element/solution element(CE/SE)method and the local discontinuous basis functions of the space-time discontinuous Galerkin(DG)finite element method,are the two key ingredients of the new *** staggered spacetime mesh is constructed using the cell-vertex structure of the underlying spatial *** universal definitions of CEs and SEs are independent of the underlying spatial mesh and thus suitable for arbitrarily unstructured *** solution within each physical time step is updated alternately at the cell level and the vertex *** this solution updating strategy and the DG ingredient,the new scheme here is termed as the discontinuous Galerkin cell-vertex scheme(DG-CVS).The high order of accuracy is achieved by employing high-order Taylor polynomials as the basis functions inside each *** present DG-CVS exhibits many advantageous features such as Riemann-solver-free,high-order accuracy,point-implicitness,compactness,and ease of handling boundary *** numerical tests including the scalar advection equations and compressible Euler equations will demonstrate the performance of the new method.

关键词： High order method space-time method cell-vertex scheme(CVS) conservation laws

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Extension of the High-Order space-time Discontinuous Galerkin Cell Vertex Scheme to Solve time Dependent Diffusion Equations

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Communications in Computational Physics 2012年第5期11卷 1503-1524页

作者： Shuangzhang Tu Gordon W.Skelton Qing Pang Department of Computer Engineering Jackson State UniversityJacksonMS 39217USA

In this paper,the high-order space-time discontinuous Galerkin cell vertex scheme(DG-CVS)developed by the authors for hyperbolic conservation laws is extended for time dependent diffusion *** the extension,the treatment of the diffusive flux is exactly the same as that for the advective *** to the Riemannsolver-free and reconstruction-free features of DG-CVS,both the advective flux and the diffusive flux are evaluated using continuous information across the cell *** a result,the resulting formulation with diffusive fluxes present is still consistent and does not need any extra ad hoc techniques to cure the common“variational crime”problem when traditional DG methods are applied to diffusion *** this reason,DG-CVS is conceptually simpler than other existing DG-typed *** numerical tests demonstrate that the convergence order based on the L_(2)-norm is optimal,i.e.O(h^(p+1))for the solution and O(h^(p))for the solution gradients,when the basis polynomials are of odd *** even-degree polynomials,the convergence order is sub-optimal for the solution and optimal for the solution *** same odd-even behaviour can also be seen in some other DG-typed methods.

关键词： High-order method space-time method discontinuous Galerkin(DG)method cellvertex scheme(CVS) diffusion equations

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