|Genuinely multidimensional schemes hyperbolic systems wave equations Euler equations evolution Galerkin schemes space-time conservative methods high order accuracy shock solutions|
In this dissertation we present two kinds of multidimensional schemes for hyperbolic systems based on triangular meshes. The first kind of schemes are evolution Galerkin schemes (EG) which are truly multidimensional schemes and the second kind is a new space-time conservative central-type method which we name a slope propagation (SP) method. Our first scheme is an extension of the EG schemes for hyperbolic systems from rectangular to triangular meshes. We develop EG schemes for the linear wave equation system, the nonlinear wave equation system, the linearized Euler equations, the advection wave equation system and the nonlinear Euler equations for structured/unstructured triangular meshes. We have also extended these EG schemes on triangular meshes to second order by using linear reconstruction. The accuracy and experimental order of convergence (EOC) of the schemes are demonstrated by numerical experiments. The accuracy of second order scheme is several times greater than the first order however the EOC of 2 has not been achieved. Several numerical test cases are presented which show that apart from such convergence difficulties EG schemes work equally well for structured and unstructured triangular meshes. Our second scheme i.e. the SP-method, which we have newly introduced is a space-time conservative second-order scheme. The scheme treats space and time in a unified manner. The flow variables and their slopes are the basic unknowns in the scheme. The scheme utilizes the advantages of the space-time conservation element and solution element (CE/SE) method (Chang, 1995) as well as central schemes (Nessyahu and Tadmor, 1990). However, unlike the CE/SE method the present scheme is Jacobian-free and hence like the central schemes can also be applied to any hyperbolic system. In Chang's method a finite difference approach is being used for the slope calculation in case of nonlinear hyperbolic equations. We propose to propagate the slopes by a scheme even in the case of nonlinear systems. By introducing a suitable limiter for the slopes of flow variables, we can apply the same scheme to linear and non-linear problems with discontinuities. The scheme is simple, efficient and has a good resolution especially at contact discontinuities. We derive the scheme for one and two space dimensions. In two-space dimensions we use structured triangular mesh. The second order accuracy of the scheme has been verified by numerical experiments. Several numerical tests presented in this dissertation validate the accuracy and robustness of the present scheme.