Nonlinear solution techniques for solving a Monge-Amp\`ere equation for redistribution of a mesh

TL;DR

This paper investigates nonlinear solution techniques for the Monge-Ampère equation used in mesh redistribution, introducing a new linearisation method that is faster, more robust, and parameter-free, with potential applications in geophysical flow simulations.

Contribution

A novel linearisation approach for the Monge-Ampère equation that simplifies computation and improves robustness compared to existing methods.

Findings

The new linearisation method is faster and more robust.

The linearised equations resemble advection-diffusion equations, enabling CFD tools.

The method shows promise for adaptive mesh redistribution in geophysical flows.

Abstract

A Monge-Amp\`ere (MA) equation arises when seeking an optimally transported mesh that equidistributes a given monitor function in Cartesian space. This MA equation is a fully nonlinear PDE, with a source term that is a function of the gradient of the solution. This nonlinear source term is an additional computational challenge that has received little attention from MA applications in other fields. There are two major components needed to find a solution to the MA equation: a spatial discretisation and an algorithm to find a solution of the resulting nonlinear algebraic equations. There have been a number of different approaches proposed in the literature to solve the MA equation but none of which perform consistent comparisons across both algorithmic and discretisation differences. In this study we explore different algorithmic methods for the MA equation all within the context of a finite volume spatial discretisation. We introduce a new linearisation of the MA equation that neglects the nonlinearities arising from the source term and show that it leads to a method that is fast, robust and free of tuning parameters. We present numerical experiments that show methods based on this linearisation of the MA equation are more computationally efficient than those that rely on other techniques such as a parabolic relaxation. Further, the equations resulting from a full linearisation of the MA equation, equivalent to using Newton's method, can be seen as analogous to an advection-diffusion equation. This allows many tools that exist for computational fluid dynamics to be re-factored easily to solve the MA equation. The robustness and efficiency of the newly introduced method gives hope that an adaptive solver for geophysical flows using mesh redistribution can be computationally feasible in the near future.