Approximate solutions to the Zakharov equations via finite differences
Journal of Computational Physics
On the representation of operators in bases of compactly supported wavelets
SIAM Journal on Numerical Analysis
Wavelets and the numerical solution of partial differential equations
Journal of Computational Physics
A conservative difference scheme for the Zhakarov equations
Journal of Computational Physics
Finite difference method for generalized Zakharov equations
Mathematics of Computation
A fast spectral algorithm for nonlinear wave equations with linear dispersion
Journal of Computational Physics
On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime
Journal of Computational Physics
Journal of Computational Physics
Numerical solution of the Gross--Pitaevskii equation for Bose--Einstein condensation
Journal of Computational Physics
SIAM Journal on Numerical Analysis
SIAM Journal on Scientific Computing
DSC time-domain solution of Maxwell's equations
Journal of Computational Physics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
An efficient and stable numerical method for the Maxwell-Dirac system
Journal of Computational Physics
Numerical simulation of a generalized Zakharov system
Journal of Computational Physics
Local spectral time splitting method for first- and second-order partial differential equations
Journal of Computational Physics
A time-splitting spectral scheme for the Maxwell-Dirac system
Journal of Computational Physics
A Time-Splitting Spectral Method for the Generalized Zakharov System in Multi-Dimensions
Journal of Scientific Computing
Local discontinuous Galerkin methods for the generalized Zakharov system
Journal of Computational Physics
Iterative Filtering Decomposition Based on Local Spectral Evolution Kernel
Journal of Scientific Computing
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We present two numerical methods for the approximation of the generalized Zakharov system (ZS). The first one is the time-splitting spectral (TSSP) method, which is explicit, time reversible, and time transverse in variant if the generalized ZS is, keeps the same decay rate of the wave energy as that in the generalized ZS, gives exact results for the plane-wave solution, and is of spectral-order accuracy in space and second-order accuracy in time. The second one is to use a local spectral method, the discrete singular convolution (DSC) for spatial derivatives and the fourth-order Runge-Kutta (RK4) for time integration, which is of high (the same as spectral)-order accuracy in space and can be applied to deal with general boundary conditions. In order to test accuracy and stability, we compare these two methods with other existing methods: Fourier pseudospectral method (FPS) and wavelet-Galerkin method (WG) for spatial derivatives combining with the RK4 for time integration, as well as the standard finite difference method (FD) for solving the ZS with a solitary-wave solution. Furthermore, extensive numerical tests are presented for plane waves, solitary-wave collisions in 1d, as well as a 2d problem of the generalized ZS. Numerical results show that TSSP and DSC are spectralorder accuracy in space and much more accurate than FD, and for stability, TSSP requires k = O(h), DSC-RK4 requires k = O(h2) for fixed acoustic speed, where k is the time step and h is the spatial mesh size.