Convergence and Order Reduction of Diagonally Implicit Runge-Kutta Schemes in the Method of Lines

摘要

Diagonally implicit Runge-Kutta discretizations of initial-boundary value problems in partial differential equations are studied. The derivation of bounds for the full discretization error under the assumption that the grid distances in space and time are independent parameters, is considered. The method of lines approach is used to exploit the B-convergence theory for Runge-Kutta schemes applied to stiff problems. Order reduction phenomena are emphasized. Numerical examples confirm theoretical results. It is shown that in partial differential equations, order reduction severely reduces the performances of higher order schemes.