Comparing Modal and Nodal Approaches of the High Order Discontinuous Galerkin Method for the Solution of Sod's Shock Tube
Alberto Costa Nogueira Junior1; Leonardo Machado Antonio2; João Lucas de Sousa Almeida1; Cláudio Alessandro de Carvalho Silva2; Renato Fernandes Cantão3
1 IBM Research Brazil; 2 Faculdade de Engenharia Mecânica - Universidade Estadual de Campinas; 3 Instituto de Matemática - Universidade Federal de São Carlos
doi:10.20906/CPS/CILAMCE2017-1114
Resumo
The Sod's shock tube is the most standard test problem to assess the accuracy of numerical schemes for solving the 1D Euler equations of compressible gas dynamics. It consists of a one-dimensional Riemann problem whose time evolution gives rise to three characteristics describing the propagation speed of different regions of the system: the rarefaction wave, the contact discontinuity and the shock discontinuity. The discontinuous Galerkin method is a very suitable technique to discretize such problem as it is capable to accurately represent the smooth regions of the problem's solution while capturing the typical jump discontinuities mentioned above. In this work, we compare the two well established versions of the high-order Discontinuous Galerkin Finite Element Method (DG-FEM), namely, modal and nodal approaches [Karniadakis & Sherwin-2005, Hestaven & Warburton-2008]. Essentially, the difference between the two formulations resides in the numerical interpolation basis functions used by each approach: Legendre polynomials for modal and Lagrange polynomials for nodal, besides the choice of quadrature points distribution: Gauss-Legendre (GL) and Gauss-Legendre Lobatto (GLL) for modal and only GLL for nodal. To ensure numerical stability along time integration, we used viscous subcell shock capturing strategy [Persson & Peraire-2006, Klockner & Warburton-2011] with two different definitions for the artificial viscosity function: (1) elementwise constant or (2) locally C0 linear. Shock detecting algorithm was set with the same parameters for both DG formulations. The numerical fluxes for the convective and viscous operators were set as the monotone Lax-Friedrichs and LDG in both cases. Nonlinear physical flux terms were treated in the same way for both discretizations using a projection operation in modal space with over integration to keep consistency and stability. An explicit fourth-order strong stability preserving Runge-Kutta time integrator was used in both approaches. Numerical experiments were devised for dif
Palavras-chave: high-order; discontinuous Galerkin; modal; nodal; viscous shock capturing; compressible gas dynamics