High Resolution Hybrid Finite Difference Scheme For Hyperbolic Of Conservation Laws

In this paper we use Taylor expansion, derive a series of semi-discrete compact schemes. The compact upwind schemes descried in this paper be demonstrated to converge for smooth solution .However, in the presence of discontinues solutions Gibbs phenomena do occur, which contaminate the solution and may lead to nonlinear instability .In order to cure this deficiency, linear algorithms must be replaced with nonlinear ones, which have a shock-capturing capability. A good example of such scheme is the so-called total variation diminishing(TVD) scheme, which probably represent the best compromise between accuracy and computation cost. The TVDtype schemes are a kind of high-resolution shock-capturing schemes. We briefly discuss the non-oscillatory and non-free-parameter dissipation difference (NND) scheme.Hybrid compact-TVD type(HCT) scheme is proposed to obtain high accurate and high resolution in discontinuous problem . In the present paper we introduce an efficient hybrid compact-NND (HCN) scheme The algorithm is based on a high order compact upwind algorithm in conservation form to solve for the smooth part of the flow field, which is coupled with a lower order TVD type scheme ,such as NND scheme, to capture the discontinuities(HCN scheme). We use a new method of flux splitting to compute nonlinear systems of conservation laws. The study also includes a comparison with compact scheme, NND scheme and hybrid scheme. Some numerical results are given.The last chapter of this paper describe the numerical method for solving liquid dynamics problem. These include analytical eigenvalues and eigenvectors of the Jacobian matrix, flux splitting and the hybrid scheme for solution of Euler equations using real gas state laws. The algorithm here developed is proven to have better resolution properties at a lower computation cost.