A modified characteristic finite element method for a fully nonlinear formulation of the semigeostrophic flow equations

TL;DR

This paper introduces a fully discrete modified characteristic finite element method for solving the nonlinear Monge-Ampère and transport equations in semigeostrophic flow models, with proven convergence and numerical validation.

Contribution

It develops a novel finite element approach for the fully nonlinear semigeostrophic equations using vanishing moment regularization, achieving optimal convergence rates.

Findings

Method converges with optimal order under mesh constraints

Error bounds explicitly depend on regularization parameter

Numerical tests confirm theoretical accuracy and efficiency

Abstract

This paper develops a fully discrete modified characteristic finite element method for a coupled system consisting of the fully nonlinear Monge-Amp\'ere equation and a transport equation. The system is the Eulerian formulation in the dual space for the B. J. Hoskins' semigeostrophic flow equations, which are widely used in meteorology to model slowly varying flows constrained by rotation and stratification. To overcome the difficulty caused by the strong nonlinearity, we first formulate (at the differential level) a vanishing moment approximation of the semigeostrophic flow equations, a methodology recently proposed by the authors \cite{Feng1,Feng2}, which involves approximating the fully nonlinear Monge-Amp\'ere equation by a family of fourth order quasilinear equations. We then construct a fully discrete modified characteristic finite element method for the regularized problem. It is shown that under certain mesh and time stepping constraints, the proposed numerical method converges with an optimal order rate of convergence. In particular, the obtained error bounds show explicit dependence on the regularization parameter $\vepsi$. Numerical tests are also presented to validate the theoretical results and to gauge the efficiency of the proposed fully discrete modified characteristic finite element method.