Grant: Discovery Grants Program - Individual

Mathematically, many models of interesting physical phenomena are written as partial differential equations (PDEs). Except for certain idealized situations, the PDEs which result are not amenable to analytical solution. Instead we rely on numerical approximations. I am particularly interested in time-dependent PDEs in two or three spatial dimensions whose solutions exhibit large solution variation, some type of singularity, blow-up or moving fronts. To efficiently track these regions of interest it is necessary to develop methods which adapt to the rapidly evolving features of the solution. Specifically, I am interested in both the practical issue of implementation and theoretical investigation of the natural marriage of two computational approaches which have received significant attention over the last 20 years - moving mesh methods and domain decomposition (in the context of Schwarz waveform relaxation). These new methods are known as Schwarz waveform relaxation moving mesh methods. The strategy works by using moving meshes to track interesting features of the solution in space while the domain decomposition and waveform relaxation allows one to isolate difficult features of the solution in time. Domain decomposition allows a problem to be broken up into small pieces. Each piece may be solved independently, ideally in parallel, and the pieces are then recombined to provide a solution for the whole problem. This strategy is now attractive given the relatively easy access to high performance computing resources. This topic will be of interest to numerical analysts as well as engineers and applied mathematicians whose work involves the simulation of complicated physical processes modeled by systems of partial differential equations.