Approximation of discontinuous solutions in high order discontinous Galerkin schemes

The paper concerns high order discontinuous Galerkin schemes. The numerical solution of ordinary differential equations is considered for those problems where the approximation of a discontinuous solution is required. It will be shown that the high order discontinuous Galerkin approximation results in solution overshoots on a grid cell which contains a discontinuity. For a linear problem, analytical expressions to evaluate the amplitude of the solution overshoot are obtained. Numerical examples confirming the theoretical results are given for both linear and nonlinear problems.