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

TL;DR

This paper introduces mass-conserving biorthogonal hierarchical basis functions for sparse grid combination techniques, improving accuracy and stability in kinetic plasma simulations by addressing conservation and numerical issues.

Contribution

The work develops two novel hierarchical basis functions that enhance mass conservation and stability in sparse grid kinetic simulations, a significant improvement over existing methods.

Findings

Mass conservation is achieved with the new basis functions.

Accuracy is significantly increased in finite-volume advection tests.

Numerical stability is improved in semi-Lagrangian Vlasov--Poisson 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. Further numerical experiments based on the combination technique applied to a semi-Lagrangian Vlasov--Poisson solver show a stabilizing effect of the new bases on the simulations.