Fast numerical treatment of nonlinear wave equations by spectral methods

Abstract

A method is presented that accelerates spectral methods for numerical solution of a broad class of nonlinear partial differential wave equations that are first order in time and that arise in plasma wave theory. The approach involves exact analytical treatment of the linear part of the wave evolution including growth and damping as well as dispersion. After introducing the method for general scalar and vector equations, we discuss and illustrate it in more detail in the context of the coupling of high-and low-frequency plasma wave modes, as modeled by the electrostatic and electromagnetic Zakharov equations in multiple dimensions. For computational efficiency, the method uses eigenvector decomposition, which is particularly advantageous when the wave damping is mode-dependent and anisotropic in wavenumber space. In this context, it is shown that the method can significantly speed up numerical integration relative to standard spectral or finite difference methods by allowing much longer time steps, especially in the limit in which the nonlinear Schrödinger equation applies.

Do Langmuir wave packets in the solar wind collapse?

“…Other wave packets analyzed during multiple strong turbulence runs with the 3D electromagnetic simulation code of Skjaeraasen et al [2011] yield almost identical results (not shown). In the earlier stages of collapse the electric fields of wave packets cannot be clearly distinguished from the background Langmuir waves; however, the radial functions of these packets tend to be better fitted by the Gaussian than the Lorentzian potential.…”

Mitigation of two-plasmon decay in direct-drive inertial confinement fusion through the manipulation of ion-acoustic and Langmuir wave damping

“…Equation 3is currently solved by neglecting collisional absorption, pump depletion, and nonlinear terms (i.e., trivially), while Eqs. 4and 5are advanced by a pseudospectral method: 62,63 All the linear propagation terms of Eqs. 4and 5are computed in Fourier space (Landau damping can be easily written in k space), while the nonlinear term in Eq.…”

Langmuir “snakes” and electrostatic decay in the solar wind

“…[7] We solve the Zakharov equations in three dimensions on a 128 3 grid with dimensions of (1200 D ) 3 , using the code in Graham et al [2011]. The numerical integration is optimized using the phase-removal scheme of Skjaeraasen et al [2011]. In our simulations, damping is applied to Langmuir waves at all k at a rate 0 = -6 10 -4 !…”

“…The numerical integration is optimized using the phase-removal scheme of Skjaeraasen et al [2011]. In our simulations, damping is applied to Langmuir waves at all k at a rate 0 = -6 10 -4 !…”