NONLINEARLY STABLE COMPACT SCHEMES FOR SHOCK CALCULATIONS

In this paper the authors discuss the applications of high-order compa ct finite difference methods for shock calculations. The main idea is the definition of a local mean that serves as a reference for introduc ing a local nonlinear limiting to control spurious numerical oscillati ons while keeping the formal accuracy of the scheme. For scalar conser vation laws, the resulting schemes can be proven total variation stabl e in one-space dimension and maximum norm stable in multispace dimensi ons. Numerical examples are shown to verify accuracy and stability of such schemes for problems containing shocks. The idea in this paper ca n also be applied to other implicit schemes such as the continuous Gal erkin finite element methods.