Grid-based solvers for the simulation of kinetic equations suffer from
the curse of dimensionality. The sparse grid combination technique is a
means to reduce the number of degrees of freedom in high dimensions,
however, the hierarchical representation for the combination step with
the state-of-the-art hat functions suffers from poor conservation
properties and numerical instability.
We introduce 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.
This is joint work with Theresa Pollinger, Johannes Rentrop und Dirk