Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes .1.

This is the first paper in a series in which a class of nonoscillatory high order accurate sell-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 nux 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.