DeepAI AI Chat
Log In Sign Up

Newton Type Methods for solving a Hasegawa-Mima Plasma Model

by   Sophie M. Moufawad, et al.

In [1], the non-linear space-time Hasegawa-Mima plasma equation is formulated as a coupled system of two linear PDE's, a solution of which is a pair (u, w). 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. In a more recent work [2], a numerical approach consisting of a finite element space-domain combined with an Euler-implicit time scheme was used to discretize the coupled variational Hasegawa-Mima model. A semi-linear version of this implicit nonlinear scheme was tested for several types of initial conditions. This semi-linear scheme proved to lack efficiency for long time, which necessitates imposing a cap on the magnitude of the solution. To circumvent this difficulty, in this paper, we use Newton-type methods (Newton, Chord and an introduced Modified Newton method) to solve numerically the fully-implicit non-linear scheme. Testing these methods in FreeFEM++ indicates significant improvements as no cap needs to be imposed for long time. In the sequel, we demonstrate the validity of these methods by proving several results, in particular the convergence of the implemented methods.


page 22

page 23

page 24

page 25


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

In a recent work, two of the authors have formulated the non-linear spac...

Convergence analysis of a numerical scheme for a tumour growth model

We consider a one-spatial dimensional tumour growth model that consists ...

Accurate Spectral Collocation Solutions to some Bratu's Type Boundary Value Problems

We solve by Chebyshev spectral collocation some genuinely nonlinear Liou...

Numerical Analysis for Real-time Nonlinear Model Predictive Control of Ethanol Steam Reformers

The utilization of renewable energy technologies, particularly hydrogen,...

IFOSMONDI Co-simulation Algorithm with Jacobian-Free Methods in PETSc

IFOSMONDI iterative algorithm for implicit co-simulation of coupled phys...