We consider the DG method to solve the hyperbolic conservation laws and study how to use the local projection limiting in the characteristic fields. Numerical experiments for 1D Riemann problem and 2D double Mach reflection problem are presented.

We investigate the performance of a hybrid compact-WENO finite difference scheme for numerical simulations of the detonation waves on uniformly discretized Cartesian domain.

In the research, we mainly use the RKDG methods based on nonuniform mesh in 1D and unstructured mesh in 2D to solve Burgers equation as an example.

Hybrid scheme is designed to solve multi-fluids Euler equation in its conservative form everywhere except at the material interfaces where the primitive form and the auxiliary advection equation are used instead.

Shallow water equation (SWE) is the hyperbolic systems of conservation laws with source terms (also called balance laws). To preserve exactly balance laws, here, we apply the well-balanced nodal DG meth-ods for 1D SWE as an example.

The solution of a hyperbolic conservation laws may have discontinuities, especially for problems containing both shocks and complicated smooth solution structures. The key idea is to detect such discontinuities using Fourier conjugate Method.