Goal Oriented Time Adaptivity Using Local Error Estimates

We consider initial value problems (IVPs) where we are interested in a quantity of interest (QoI) that is the integral in time of a functional of the solution. For these, we analyze goal oriented time adaptive methods that use only local error estimates. A local error estimate and timestep controller for step-wise contributions to the QoI are derived. We prove convergence of the error in the QoI for tolerance to zero under a controllability assumption. By analyzing global error propagation with respect to the QoI, we can identify possible issues and make performance predictions. Numerical tests verify these results. We compare performance with classical local error based time-adaptivity and a posteriori based adaptivity using the dual-weighted residual (DWR) method. For dissipative problems, local error based methods show better performance than DWR and the goal oriented method shows good results in most examples, with significant speedups in some cases.

SOME LOW ORDER TWO-STEP ALMOST P-STABLE METHODS WITH PHASE-LAG OF ORDER INFINITY FOR THE NUMERICAL INTEGRATION OF THE RADIAL SCHRÖDINGER EQUATION

Some two-step P-stable methods with phase-lag of order infinity are developed for the numerical integration of the radial Schrödinger equation. The methods are of O(h2) and O(h4) respectively. We produce, based on these methods and on a new local error estimate, a very simple variable step procedure. Extensive numerical testing indicates that these new methods are generally more accurate than other two-step methods with higher algebraic order.