Devising discontinuous Galerkin methods for non-linear hyperbolic conversation laws
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
The Numerical Study of Singular Shocks Regularized by Small Viscosity
Journal of Scientific Computing
Discontinuous Galerkin Methods: Theory, Computation and Applications
Discontinuous Galerkin Methods: Theory, Computation and Applications
Numerical approximations of Riemann solutions to multiphase flows used in petroleum engineering
MACMESE'07 Proceedings of the 9th WSEAS international conference on Mathematical and computational methods in science and engineering
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We present a numerical study for two systems of conservation laws using a spacetime discontinuous Galerkin (SDG) method with causal spacetime triangulations and the piecewise constant Galerkin basis. The SDG method is consistent with the weak formulation of conservation laws, and, in the case of strictly hyperbolic systems, also with the Lax entropy condition. Convergence of the method was shown for a special class of hyperbolic systems (Temple systems).The initial data we consider lead to nonclassical shocks. The first part of our study is for the Keyfitz-Kranzer system. We compute the SDG solutions approximating overcompressive and singular shocks, and note that our results are consistent with those obtained by [Sanders, and Sever 2003] using a finite difference scheme. The second system we consider is an approximation of a three-phase flow in the petroleum reservoirs. Numerical solutions for this system were computed by [Schecter, Plohr, and Marchesin 2004] using the Dafermos regularization and a technique for numerical solving of ordinary differential equations. We compute the SDG approximation to a solution containing a transitional shock.We note that even though convergence of the SDG method was shown so far only for Temple systems, numerical examples herewith show that it can be used successfully in approximating solutions of more general conservation laws.