Parabolic Monge-Ampère methods for blow-up problems in several spatial dimensions

This paper constructs and analyses an adaptive moving mesh scheme for the numerical simulation of singular PDEs in one or more spatial dimensions. The scheme is based on computing a Legendre transformation from a regular to a spatially non-uniform mesh via the solution of a relaxed form of the Monge–Ampère equation. The method is shown to preserve the inherent scaling properties of the PDE and to identify natural computational coordinates. Numerical examples are presented in one and two dimensions which demonstrate the effectiveness of this approach.