In Zhang and Shu (J. Comput. Phys. 229:3091–3120, 2010), two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations in Zhang and Shu (J. Comput. Phys. 229:8918–8934, 2010). The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.
Hyperbolic conservation laws Finite volume scheme Discontinuous Galerkin method Essentially non-oscillatory scheme Weighted essentially non-oscillatory scheme Maximum principle Positivity preserving High order accuracy Strong stability preserving time discretization Passive convection equation Incompressible flow Compressible Euler equations Triangular meshes
Liu, W., Cheng, J., Shu, C.-W.: High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations. J. Comput. Phys. 228, 8872–8891 (2009)
Zhang, X., Shu, C.-W.: On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918–8934 (2010)