Application of Multiscale Techniques to Hyperbolic Conservation Laws

The solution of hyperbolic conservation laws is characterized by the nite speed of propagation and the development of disconti-nuities. Stable and robust numerical schemes, e.g. nite volume schemes, are based on upwind techniques in order to handle shocks and contact discontinuities. In general, schemes of this type are very expensive. In 1993, Harten derived a concept to reduce the computational costs. It is based on a multiscale decomposition such that regions with singularities can be located. Near discontinuities, an expensive upwind scheme is applied ; otherwise a much cheaper nite diierence scheme of high order is used. Harten's multiscale decomposition is based on the primitive function of the solution. This approach is inherently restricted to structured grids. In this paper we present an alternative multiscale decomposition using general reconstruction methods. Here, we consider the one{dimensional case and verify that our concept coincides with those of Harten's decomposition method. Furthermore, we apply our algorithm to dimensionally reduced hypersonic stagnation point ows around spheres. All these techniques can be extended to multidimensional problems even for unstruc-tured grids (see 7]). x1. Introduction In the present paper, we consider partial diierential equations of the form @ u(t; x) @ t + @ f(u(t; x)) @ x = s(x; u(t; x)); 0 < t < T; a < x < b (1:1) with T > 0 and a; b 2 IR xed, so{called conservation laws. Herein, u : a; b] ! IR is the conservative quantity, f : IR ! IR the ux and s : IR ! IR the source term. In order to ensure a unique solution, an initial condition u(0; x) = u 0 (x); a < x < b (1:2) has to be imposed. In practical applications, boundary conditions of the form g a (u(t; a); u x (t; a)) = 0; g b (u(t; b); u x (t; b)) = 0; 0 t T (1:3) are also needed.