Accuracy optimized methods for constrained numerical solutions of hyperbolic conservation laws (Q1316040)

A new method is presented for approximating the solution of scalar hyperbolic conservation laws. The central idea of this is to construct a general method, named AOM, which optimizes the accuracy of the approximation subject to constraints such as monotonicity preserving, total variation diminishing or entropy constraints. This requires to solve an additional optimization problem. But this is necessary only in those narrow regions where an unmodified method would violate imposed constraints. Such constraints lead to a well-posed quadratic programming problem. In this paper the linear advection problem and the specific nonlinear Burgers equation are solved with the help of second-order difference methods. Some numerical examples are listed and compared with the TVD methods presented by \textit{P. K. Sweby} [SIAM J. Numer. Anal. 21, 995-1011 (1984; Zbl 0565.65048)].