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

This is the first paper in a series in which 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 are constructed and analyzed. In this paper a scheme which is of third order of accuracy in the sense of flux approximation is presented, using scalar one-dimensional initial value problems as a model. For this model, the schemes are made to satisfy a local maximum principle and a nonoscillatory property. The method uses a simple centered stencil with quadratic reconstruction followed by two modifications, imposed as needed. The first enforces a local maximum principle; the second guarantees that no new extrema develop. The schemes are self similar in the sense that the numerical flux does not depend explicitly on the grid size, i.e., there are no grid size dependent limits involving free parameters as in, e.g., [Math. Comp., 49 (1987), pp. 105–121, Math. Comp., 49 (1987), pp. 123–134, Math. Comp., 52 (1989), pp. 411–435, J. Comp. Phys., 84 (1989), pp. 90–113]. Combining the nonoscillatory property and the local maximum principle TVB (total variation bounded) property is achieved. Hence convergence of a subsequence of the numerical solutions is obtained as the step size approaches zero. Numerical results are encouraging. Extensions to systems and/or higher dimensions will appear in future papers, as will extensions to higher orders of accuracy.