Computation of nonclassical shocks using a spacetime discontinuous galerkin method

Abstract

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 successfully used in approximating solutions of more general conservation laws.