High Order Accuracy Optimized Methods for Constrained Numerical Solutions of Hyperbolic Conservation Laws

• Published in: SIAM journal on scientific computing Vol. 15; no. 4; pp. 846 - 865
• Main Authors: Coray, C., Koebbe, J.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.07.1994
• Subjects: Accuracy; Applied mathematics; Approximation; Conservation laws; Methods; Numerical analysis; Optimization
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/0915052

A high order accuracy generalization of accuracy optimized methods (AOMs) for the numerical solution of scalar hyperbolic conservation is described. This process includes a presentation of a general framework for the construction of high order accurate base schemes that are linearly stable, consistent, and conservative. The AOM defines and solves a quadratic programming problem at each discrete time level to minimize perturbations from the high order base schemes subject to imposed constraints. The constraints are used to imposed desired behavior on the numerical approximation to the solution of the conservation law. The resulting schemes retain the high order accuracy of the base scheme away from shocks, and minimally perturb the high order base schemes locally where necessary to meet the imposed constraints. The resulting schemes compare favorably with other high resolution schemes for scalar conservation laws. Numerical examples are presented to illustrate convergence rates for the high order methods, stability regions, and AOM results for linear advection of discontinuities and development and transport of shocks in Burgers' equation. The constraints used in this work lead to a systematic method for construction of high order accurate total variation diminishing (TVD) schemes.