On some optimization procedures for shock localization

The shock wave localization accuracy is investigated for the Miranker-Pironneau method and for some of its modifications by means of methods of classical variational calculus. The Miranker-Pironneau method may be difficult to implement when the entropy condition is included in the formulation of an optimization problem, as an active constraint. In this connection, an alternative optimization problem using artificial viscosity in the basic functional, is considered. A problem of the motion of a stationary shock wave in a gas is considered in the numerical solution of one-dimensional gasdynamic equations in Eulerian variables. It is shown that the application of such a functional yields a trajectory which coincides with the true discontinuity trajectory in the case of a shock wave moving at a constant speed. An algorithm in which the shock localization is based on the minimization of the univariate function is also proposed. It is shown that the use of the functional and the univariate function yields more accurate localization results as compared to the original Miranker-Pironneau method.