Spatial Manifestations of Order Reduction, and Overcoming it via Weak Stage Order
Benjamin Seibold (Mathematics, Temple University)
Order reduction, i.e., the convergence of the solution at a lower rate than the formal order of the chosen time-stepping scheme, is a fundamental problem in stiff ODEs, and particularly in PDE IBVPs. Runge-Kutta schemes with high stage order provide a remedy, but unfortunately high stage order is incompatible with diagonally-implicit (DIRK) schemes. We first highlight the spatial manifestations of order reduction in PDE IBVPs. Then we introduce the concept of weak stage order, and demonstrate (a) how it can overcome order reduction in important linear PDE problems; and (b) that it can recover high-order convergence with DIRK schemes.