A mixed Fourier-variational approach to solve differential or integro-differential wave equations for magnetised plasmas

Abstract

The All ORders Spectral Algorithm (AORSA) wave equation solver by Jaeger (Jaeger et al 2001 Phys. Plasmas 8 1573) solves the integro-differential wave equation relevant for the radio frequency (RF) domain and for fusion-relevant conditions in tokamaks or stellarators, retaining all finite Larmor radius corrections by substituting the continuous Fourier integrals by a sum over a discrete set of modes. Its strength is also its weakness: the simplicity of the method results in significant computational effort, a full matrix needing to be inverted to solve the associated linear system. Based on the notion that modes are gradually more independent if their eigenvalues differ, the present paper proposes a straightforward numerical method to partly alleviate this need, allowing to substitute the full system matrix by a banded one. The adopted method can be applied to a wide variety of equations. A few 1D examples—of relevance for solving the wave equation in the RF domain of frequencies—are provided: the tunneling equation is used to illustrate the potential of the method, and the all-FLR wave equation (retaining all Finite Larmor Radius corrections in the dielectric response) adopted by Jaeger is solved comparing the solutions found to those based on simpler models (a cold plasma and a ‘tepid plasma’ - i.e. a kinetic model truncated at zero order in Larmor radius—description).