An implicit method for hyperbolic conservation laws on meshes with small cells in one dimension

In this paper we study a new method for solving hyperbolic conservation laws on a Cartesian mesh with some small cells. Our main task here is to devise a stable algorithm in the small zones. An algorithm proposed by Berger and LeVeque combines the small zones with neighboring zones and solves rotated Riemann problems. This method is very geometrically oriented. It requires knowledge of the areas of the small cells, as well as the areas of cells contained in various {open_quote}boxes{close_quotes} drawn from the edges of the small cells. Here we propose a more algebraic algorithm: we combine an implicit method with an explicit second-order conservative finite difference scheme. In section 2 the basic algorithm in one dimension is presented, as well as the slopes calculations and the iterative procedure. In section 3 we present some numerical results for the 1-D advection equation and the inviscid Burger`s equation