A parallel implementation of a two-dimensional fluid laser–plasma integrator for stratified plasma–vacuum systems

Abstract

A two-dimensional fluid laser-plasma integrator for stratified plasma-vacuum systems is presented. Inside a plasma, a laser pulse can be longitudinally compressed from ten or more wave lengths to one or two cycles. However, for physically realistic simulations, transversal effects have to be included, because transversal instabilities can destroy the pulse and transversal compression in the plasma as well as focusing in vacuum allows much higher intensities to be reached. In contrast to the one-dimensional case, where a two-step implementation of the Gautschi-type exponential integrator with constant step size turned out to be sufficient, it is essential to enable changes of the time step-size for the two-dimensional case. The use of a one-step version of the Gautschi-type integrator, being accurate of second order independent of the highest frequencies arising in the system, is proposed. In vacuum this allows to take arbitrarily large time steps. To optimize runtime and memory requirements within the plasma, a splitting of the Laplacian is suggested. This splitting allows to evaluate the matrix functions arising in the Gautschi-type method by onedimensional Fourier transforms. It is also demonstrated how the different variants of the scheme can be parallelized. Numerical experiments illustrate the superior performance and accuracy of the integrator compared to the standard leap-frog method. Finally, we discuss the simulation of a layered plasma vacuum structure using the new method.

Exponential integrators

“…For the time integration, the scheme exp4 by Hochbruck et al (1998), which is a particular variant of the exponential Rosenbrock-type method discussed in Example 2.25, is used. Karle, Schweitzer, Hochbruck, Laedke and Spatschek (2006) and Karle, Schweitzer, Hochbruck and Spatschek (2008) suggest using Gautschi-type integrators for the simulation of nonlinear wave motion in dispersive media. The model derived in these papers applies to laser propagation in a relativistic plasma.…”

Rank-adaptive dynamical low-rank integrators for first-order and second-order matrix differential equations

“…Rank reduction is disabled in this phase. Then let r * denote the number of singular values of A ν greater than or equal to tol ν defined in (15). If r * < r 1 , we continue the integration with r ν+1 = r * .…”

“…As an example for second-order problems, we consider a reduced model of laserplasma interaction from [14,15,26]. It is given by a wave equation with spacedependent cubic nonlinearity on a bounded, rectangular domain given in (16) with periodic boundary conditions.…”