We present a Lagrangian scheme for one-dimensional Euler equations.The scheme uses Runge-Kutta discontinuous Galerkin (RKDG) finite element method to solve Euler equations under Lagrangian framework.The mesh moves with fluid flow.The scheme is conservative for density,momentum and total energy.It achieves second-order accuracy both in space and time.Numerical tests are presented to demonstrate accuracy and non-oscillatory properties of the scheme.

We present a Lax-Wendroff discontinuous Galerkin (LWDG) method combining with adaptive mesh refinement (AMR) to solve three-dimensional hyperbolic conservation laws. Compared with Runge-Kutta discontinuous finite element method (RKDG) the method has higher efficiency. We give an effective adaptive strategie. Equidistribution strategy is easily implemented on nonconforming tetrahedral mesh. Error indicator is introduced to solve three-dimensional Euler equations. Numerical experiments demonstrate that the method has satisfied numerical efficiency.

Four semi-implicit discretization schemes are used to construct preconditioners.And preconditioned Jacobian-free Newton-Krylov (JFNK) are presented to solve one-dimensional problems.Numerical results show that the preconditioning methods improve the convergence behavior of JFNK method dramatically.

A numerical method is developed for two-dimensional nonequilibrium radiation diffusion equations.Discontinuous Galerkin method is applied in spatial diseretization in which numerical flux is constructed with weighted flux averages.Implicit-explicit integration factor method for time discretization is applied to nonlinear ordinary differential equations which is obtained with discontinuous Galerkin method. Radiation diffusion equations with multiple materials are solved on unstructured grids in numerical tests.It demonstrates that the method is effective for high nonlinear and tightly coupled radiation diffusion equations.

We discuss numerical simulation of one-dimensional non-equilibrium radiation diffusion equations.A weighted numerical flux between adjacent grid cells is obtained by solving heat conduction equation with discontinuous coefficient.With this numerical flux of diffusive generalized Riemann problem(dGRP),a discontinuous finite element method is proposed for radiation diffusion equations. A backward Euler time diseretization is applied for semi-discrete form and a Picard iteration is used to solve nonlinear system of equations.Numerical results demonstrate that the method has a capability of capturing strong gradients and can be accommodated to discontinuous diffusion coefficient.