A Finite-Element Model for the Hasegawa-Mima Wave Equation

TL;DR

This paper develops a finite element method for the Hasegawa-Mima wave equation, providing a stable semi-discretization approach that guarantees global solutions and efficient numerical tests for nonlinear systems.

Contribution

It introduces a finite element space-domain approach for the coupled Hasegawa-Mima model, ensuring global existence and practical implementation with implicit time schemes.

Findings

Global existence of solutions in $H^2$ for any time interval.

Efficient semi-discrete and full-discrete algorithms based on finite elements.

Numerical tests demonstrate the approach's effectiveness for various initial data.

Abstract

In a recent work, two of the authors have formulated the non-linear space-time Hasegawa-Mima plasma equation as a coupled system of two linear PDEs, a solution of which is a pair $(u,w)$, with $w=(I-\Delta)u$. The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa-Mima problem with periodic boundary conditions were also derived. Using the Fourier basis in the space variables, existence of solutions were obtained. Implementation of algorithms based on Fourier series leads to systems of dense matrices. In this paper, we use a finite element space-domain approach to semi-discretize the coupled variational Hasegawa-Mima model, obtaining global existence of solutions in $H^2$ on any time interval $[0,T]$ for all T. In the sequel, full-discretization using an implicit time scheme on the semi-discretized system leads to a nonlinear full space-time discrete system with a nonrestrictive condition on the time step. Tests on a semi-linear version of the implicit nonlinear full-discrete system are conducted for several initial data, assessing the efficiency of our approach.