Approximation of PDEs with Underlying Continuity Equations

Abstract

We develop a numerical method for the solution of special partial differential equations. We use an approximation space, which automatically adapts in space and time to the function that has to be approximated. For that purpose, we use the corresponding probability density function, transport maps to its probability distribution and the underlying continuity equation. The theory and numerical examples will be presented using the Schrödinger equation as the showcase PDE.