A finite-element model for the Hasegawa–Mima wave equation

Karakazian, Hagop; Moufawad, Sophie; Nassif, Nabil

doi:10.1016/j.amc.2021.126550

Cited by 3 publications

(10 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To remedy the ill-behaviored semi-linear approach (8), we propose in this paper Newton-type algorithms that are based on the well-known Newton's method for solving the full discrete system (6). System ( 6) is equivalent to finding (U, W ) ∈ R N × R N such that:…”

Section: Resultsmentioning

confidence: 99%

“…In this section, we discuss in details Newton's method (section 3.2), for solving the full discrete system (6), by first deriving the corresponding Jacobian matrix (section 3.1). Then, in sections 3.3 and 3.4, we prove respectively the existence of a unique solution to (30), and the convergence of Newton's method.…”

Section: Newton' Methodsmentioning

confidence: 99%

“…The nonlinearity of the problem originates from S(U ), that must be computed at each iteration. The derivation of S(U ) and R is detailed in [6], where they have the same block sparsity patterns as that of M and K. First results to solve (6) where obtained in [2] using a simple semi-linear practical approach:…”

Section: The Non-linear Algebraic Systemmentioning

confidence: 99%

“…U t,0 = U t ; W t,0 = W t ; error = 1; k = 0 U t+1 = U t,k ; W t+1 = W t,k ; 12: end for Algorithm 1 Solving HM using Newton's Method Input: A: stiffness matrix ; M : Mass matrix; K = M + A; S(U ); R: As defined in [6]; B(W ): Matrix (25); U 0 ; W 0 : the discrete initial condition vectors; T : end time; τ : time step; N :total number of mesh nodes; k max : maximum Newton iterations; tol: Newton's relative error stopping tolerance. Output: U : N × (T /τ + 1) matrix containing the computed vectors U t for t = 0, τ, 2τ, • • • , T 1: for t = 0 : τ : T do 2:…”

Section: Computer Simulations and Testingsmentioning

confidence: 99%

“…where V (u) = −u y i + u x j is a divergence-free vector field (div( V (u)) = 0). The full discretization of the coupled system were obtained in [2] where starting with the given initial condition at t = 0, the subsequent solutions are approximated for a chosen time interval τ , to reach the end time T in a finite number of steps. These can be sumarized in the following three sections.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Newton Type Methods for solving a Hasegawa-Mima Plasma Model

Moufawad¹,

Nassif²

2022

Preprint

View full text Add to dashboard Cite

In [1], the non-linear space-time Hasegawa-Mima plasma equation is formulated as a coupled system of two linear PDEs, a solution of which is a pair (u, w), with w = (I −∆)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. 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.

show abstract

Section: Resultsmentioning

confidence: 99%

Section: Newton' Methodsmentioning

confidence: 99%