Adaptive d-dimensional finite difference methods

Abstract

The finite difference method is a numerical approach to approximate the solution to initial boundary value problems. In my thesis I will endeavour to solve higher dimensional initial boundary value problems using a generalized finite difference method. Challenges for higher dimensional cases are at first glance the exponential increase in memory and computation. But even before those the first hurdle is finding fitting definitions and notations for the structures arising in these computations. Doing so enables the analysis and efficient implementation of approximation methods over conventional uniform grids and further extension by implementing an adaptive approach with increased efficiency in approximating the solution of - for example - the fifth dimensional Burgers' equation.