Efficient and accurate SAV schemes for the generalized Zakharov systems
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, different situations are considered and the authors draw pictures of the solutions.
scientific article; zbMATH DE number 7350009