Uniformly high order accurate essentially non-oscillatory schemes, 111
Journal of Computational Physics
Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Total-variation-diminishing time discretizations
SIAM Journal on Scientific and Statistical Computing
Weighted essentially non-oscillatory schemes
Journal of Computational Physics
Efficient implementation of weighted ENO schemes
Journal of Computational Physics
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
A high-order discontinuous Galerkin method for 2D incompressible flows
Journal of Computational Physics
Numerical Simulation of High Mach Number Astrophysical Jets with Radiative Cooling
Journal of Scientific Computing
High Order Strong Stability Preserving Time Discretizations
Journal of Scientific Computing
Journal of Computational Physics
On maximum-principle-satisfying high order schemes for scalar conservation laws
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Positivity-preserving high order finite difference WENO schemes for compressible Euler equations
Journal of Computational Physics
SIAM Journal on Scientific Computing
Positivity-preserving schemes for Euler equations: Sharp and practical CFL conditions
Journal of Computational Physics
Journal of Computational Physics
Journal of Scientific Computing
A HLL-Rankine-Hugoniot Riemann solver for complex non-linear hyperbolic problems
Journal of Computational Physics
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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.