Convergence and Order Reduction of Runge-Kutta Schemes Applied to Evolutionary Problems in Partial Differential Equations

摘要

Convergence of fully discrete explicit Runge-Kutta approximations is discussed. It is proved that under certain conditions, the order in time of the fully discrete scheme equals the conventional order of the Runge-Kutta formula being used. However, these conditions, which are necessary for the result to hold, are not natural. As a result, in many problems the order in time is strictly smaller than the conventional one, a phenomenon called order reduction. Although results are valid for parabolic and hyperbolic problems, the examples are taken from the hyperbolic field, as there explicit discretizations are most appealing.