A stable and mass-conserving sparse grid combination technique with biorthogonal hierarchical basis functions for kinetic simulations

Abstract

The exact numerical simulation of plasma turbulence is one of the assets and challenges in fusion research. For grid-based solvers, sufficiently fine resolutions are often unattainable due to the curse of dimensionality. The sparse grid combination technique provides the means to alleviate the curse of dimensionality for kinetic simulations. However, the hierarchical representation for the combination step with the state-of-the-art hat functions suffers from poor conservation properties and numerical instability.The present work introduces two new variants of hierarchical multiscale basis functions for use with the combination technique: the biorthogonal and full weighting bases. The new basis functions conserve the total mass and are shown to significantly increase accuracy for a finite-volume solution of constant advection. Numerical analysis of the new basis functions reveals that their higher dual regularity does not only lead to conservation, but also yields an L2-stable basis for the combination technique. Accordingly, further numerical experiments applying the combination technique to a semi-Lagrangian Vlasov–Poisson solver in six dimensions show a stabilizing effect of the biorthogonal and full weighting bases on the simulations.