Moving Mesh Generation Using the Parabolic Monge-Ampère Equation

Authors:
C. J. Budd;J. F. Williams
Affiliations:
mascjb@bath.ac.uk;jfw@math.sfu.ca
Venue:
SIAM Journal on Scientific Computing
Year:
2009

Citing 0
Cited 7

Fast finite difference solvers for singular solutions of the elliptic Monge-Ampère equation

Journal of Computational Physics
Numerical Simulation of Diffusive and Aggregation Phenomena in Nonlinear Continuity Equations by Evolving Diffeomorphisms

SIAM Journal on Scientific Computing
Optimal mass transport for higher dimensional adaptive grid generation

Journal of Computational Physics
Convergent Finite Difference Solvers for Viscosity Solutions of the Elliptic Monge-Ampère Equation in Dimensions Two and Higher

SIAM Journal on Numerical Analysis
Monge-Ampére based moving mesh methods for numerical weather prediction, with applications to the Eady problem

Journal of Computational Physics
Numerical solution of the Optimal Transportation problem using the Monge-Ampère equation

Journal of Computational Physics
Finite element methods for fully nonlinear second order PDEs based on a discrete Hessian with applications to the Monge-Ampère equation

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	0.02

Visualization

Abstract

This article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations (PDEs) in several spatial dimensions. The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation. This method gives a new technique for performing $r$-adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a (time-varying) scalar monitor function (used successfully in moving mesh methods in one-dimension). Detailed analysis of this new method is presented in which the convergence, regularity, and stability of the mesh is studied. Additionally, this new method is shown to be straightforward to program and implement, requiring the solution of only one simple scalar time-dependent equation in arbitrary dimension, with adaptivity along the boundaries handled automatically. We discuss three preexisting methods in the context of this work. Examples are presented in which either the monitor function is prescribed in advance, or it is given by the solution of a partial differential equation.