Stable Dynamical Adaptive Mesh Refinement

Abstract

We consider accurate and stable interpolation procedures for numerical simulations utilizing time dependent adaptive meshes. The interpolation of numerical solution values between meshes is considered as a transmission problem with respect to the underlying semi-discretized equations, and a theoretical framework using inner product preserving operators is developed, which allows for both explicit and implicit implementations. The theory is supplemented with numerical experiments demonstrating practical benefits of the new stable framework. For this purpose, new interpolation operators have been designed to be used with multi-block finite difference schemes involving non-collocated, moving interfaces.

An efficient hybrid method for uncertainty quantification

“…To extend the technique to multiple stochastic dimensions, is in principle straightforward, but some technical problems requires further research. One has to construct stable interface couplings in multiple dimensions, which can be done following the guidelines in [6,7,9].…”