Nonlinearly stable compact schemes for shock calculations

In this paper the authors discuss the applications of high-order compact finite difference methods for shock calculations. The main idea is the definition of a local mean that serves as a reference for introducing a local nonlinear limiting to control spurious numerical oscillations while keeping the formal accuracy of the scheme. For scalar conservation laws, the resulting schemes can be proven total variation stable in one-space dimension and maximum norm stable in multispace dimensions. The idea in this paper can also be applied to other implicit schemes such as the continuous Galerkin finite element methods.