Efficient and accurate SAV schemes for the generalized Zakharov systems
type: Item
instance of: scholarly article
title: Efficient and accurate SAV schemes for the generalized Zakharov systems
zbMATH Open document ID: 1465.35362
author: Yanyan Li
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2021-05-20T00:00:00Z
review text: The authors propose a numerical scheme for the generalized Zakharov or generalized vector Zakharov systems which describe the propagation of Langmuir waves in a plasma. The generalized Zakharov system is written as \( i\partial _{t}E+\Delta E-\alpha NE+\lambda \left\vert E\right\vert ^{2}E=0\), \(N_{t}=-\nabla \cdot \mathbf{V}\), \(\mathbf{V}_{t}=\frac{1}{\epsilon ^{2}} \nabla (N-\nu \left\vert E\right\vert ^{2})\), and is posed in \(\mathbb{R} ^{d}\times (0,\infty )\), \(d=1,2,3\). Here \(E\) is the envelope of the electric field, \(N\) is the deviation of the ion density from its equilibrium value, \(\epsilon \) is inversely proportional to the acoustic speed, and \( \alpha >0,\lambda ,\nu \) are real parameters. Initial conditions are imposed to \(E\), \(N\), and \(N_{t}\). The generalized vector Zakharov system is written as \(i\partial _{t}\mathbf{E}+\Delta \mathbf{E}+(1-a)\nabla (\nabla \cdot \mathbf{E})-\alpha N\mathbf{E}+\lambda \left\vert \mathbf{E}\right\vert ^{2} \mathbf{E}=0\), \(N_{t}=-\nabla \cdot \mathbf{V}\), \(\mathbf{V}_{t}=\frac{1}{ \epsilon ^{2}}\nabla (N-\nu \left\vert \mathbf{E}\right\vert ^{2})\). The authors observe that these systems conserve associated Hamiltonians. The main purpose of the paper is to propose scalar auxiliary variable schemes for these problems, which are based on the semi-implicit Crank-Nicolson-Adams-Bashforth method. The main result of the paper proves that these numerical schemes unconditionally preserve modified Hamiltonians and admit solutions if \(\alpha /\nu <0\) in the scalar case and \(a>0\) and \( \alpha /\nu <0\) in the vector case. The paper ends with the presentation and discussions of results concerning different cases: standard Zakharov system with a solitary wave-solution, 2D generalized Zakharov system with a linear damping term, soliton-soliton collisions in the 1D generalized Zakharov system, and the authors finally propose a dynamic simulation of a 3D generalized vector Zakharov system. In each case, di…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 65N35
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 76X05
zbMATH DE Number: 7350009
zbMATH Keywords: generalized Zakharov system
zbMATH Keywords: scalar auxiliary variable
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: semi-implicit scheme
zbMATH Keywords: stability
zbMATH Keywords: existence of solution
MaRDI profile type: Publication
full work available at URL: dcdsb.2020262
OpenAlex ID: W3082723250
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DOI: 10.3934/DCDSB.2020262
description: scientific article; zbMATH DE number 7350009