Efficient continuous Runge-Kutta methods for asymptotically correct defect control

Abstract

Mono-Implicit Runge-Kutta (MIRK) methods and continuous MIRK (CMIRK) methods, are used in the numerical solution of boundary value ordinary differential equations (ODEs). One way of assessing the quality of the numerical solution is to estimate its maximum defect, which is the amount by which the solution fails to satisfy the ODE. The standard approach is to perform two point sampling of the defect on each subinterval of a mesh which partitions the problem domain to estimate the maximum defect. However, the location of the maximum defect on each subinterval typically varies from subinterval to subinterval, and from problem to problem. Thus sampling at only two points typically leads to an underestimate of the maximum defect. In this thesis, we will derive a new class of CMIRK interpolants for which the location of the maximum defect on each subinterval is the same over all subintervals and problems.