A finite element method for conservation laws: Multi-dimensional case.

摘要

The RKDG-methods designed by Cockburn and Shu are new numerical methods derived to obtain approximations of the physically relevant solution of the initial boundary value problem associated with the hyperbolic conservation law(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}eqalign{lcub}&partialsb tu+div {lcub}bf f{rcub} = 0,cr&u(t = 0) = usb0,cr&u = gamma,cr{rcub}qquadeqalign{lcub}&{lcub}rm in{rcub}quad (0,T) times Omega,cr&{lcub}rm in{rcub}quad Omega,cr&{lcub}rm in{rcub}quad (0, T) times partialOmega,cr{rcub}{dollar}{dollar}(TABLE/EQUATION ENDS)where {dollar}Omegain Rsp d, u = (usb1,...,usb m)sp t{dollar}, and f is such that any real combination of the Jacobian matrices{dollar}{dollar}{lcub}sumlimitssbsp{lcub}i=1{rcub}{lcub}d{rcub}{rcub} {lcub}partial fsb iover partial u{rcub}{dollar}{dollar}has m real eigenvalues and a complete set of eigenvectors.; The study of this scheme in the multidimensional scalar case {dollar}(m = 1, d > 1){dollar} along with numerical experiments using triangular mesh and {dollar}Psp1{dollar} elements was introduced in (8).; In this thesis, we extend the work of (8). A projection different from the one in (8) will be analyzed. A maximum principle will be obtained. In section 8, a TVB property is proven. Finally, the numerical experiments especially those using {dollar}Psp2{dollar} elements will be displayed in section 9.