Adaptive spatial and temporal integrators for heat conduction problems

Abstract

We explore temporal and spatial adaptivity for approximating solutions to differential equations. We discuss temporal adaptivity applied to Runge-Kutta in time and finite-element in space approximations to solutions of time-dependent partial differential equations under the method-of-lines framework. For heat conduction problems, we discover that using an adaptive time-stepper more efficiently models the transition to the steady state than fixed time-steppers. We also measure the quality of adaptive step size controllers, implementing a bisection algorithm that finds the largest acceptable step size for a given tolerance. We also explore spatial adaptivity for finite-element solutions of partial differential equations. We find that physical problems can be resolved through high-resolution uniform grids or smaller grids better adapted to the problem. We compare both approaches by implementing a moving spatial grid through the curvilinear coordinate system, allowing us to map between spatial and computational grids. We then explore the parameters of the monitor function, which shifts the mesh points toward regions of either high curvature or high arc length.