Nonoscillatory High Order Accurate Self-similar Maximum Principle Satisfying Shock Capturing Schemes I

This is the rst paper in a series in which we construct and analyze a class of nonoscillatory high order accurate self-similar local maximum principle satisfying ( in scalar conservation law ) shock capturing schemes for solving multidimensional systems of conservation laws. In this paper we present a scheme which is of 3rd order of accuracy in the sense of ux approximation, using scalar one-dimensional initial value problems as a model. For this model, we make the schemes satisfy a local maximum principle and a nonoscillatory property. The method uses a simple centered stencil with quadratic reconstruction followed by two modi cations, imposed as needed. The rst enforces a local maximum principle, the second guarantees that no new extrema develop. The schemes are self-similar in the sense that the numerical ux does not depend explicitly on the grid size, i.e., there are no grid size dependent limits involving free parameters as in, e.g., [12],[13],[14],[15]. Combining the nonoscillatory property and the local maximum principle we achieve TVB ( Total Variation Bounded ) property. Hence we obtain convergence of a subsequence of the numerical solutions as the step size approaches zero. Numerical results are encouraging. Extensions to systems and/or higher dimensions will appear in future papers, as well as extensions to higher orders of accuracy. 3