On the Finite Difference Schemes for Burgers Equation Solution

Abstract

The new method of the difference schemes for solving the Burgers equation constructing is discovered. The method is based on the two different divergent forms for the Hopf equation. To search for the optimal difference schemes in this family, an analysis in the space of insufficient coefficients was applied using the technique of self-dual problems of linear programming solving. The multiple before the third derivate of exact solution grid mapping is used as a target functional. It is necessary and sufficient that the complementary slackness conditions be fulfilled. Based on complementary slackness conditions analysis, a new version of the Lax–Wendroff scheme is built. The new hybrid schemes with maximum anti-dispersion are also constructed. Some numerical results demonstrate the properties of the new difference schemes. Such a consideration opens up a way to build the optimal hybrid schemes with a successful choice of the target functional linear by the scheme insufficient coefficients.KeywordsBurgers equationHopf equation divergent formsFinite differenceInsufficient coefficientsLinear programmingDual problemComplementary slackness conditions