Local cubic spline interpolation for Vlasov-type equations on a multi-patch geometry

TL;DR

This paper introduces a semi-Lagrangian numerical method utilizing local cubic spline interpolation on multi-patch meshes to solve Vlasov-type equations, accommodating non-uniform and non-conforming meshes with validated results.

Contribution

It develops a novel spline-based interpolation approach for Vlasov equations on complex geometries, including derivative adaptation and global/local system solutions.

Findings

Method effectively handles non-uniform meshes.

No constraints on patch point numbers in conforming case.

Validated with 2D guiding-center model.

Abstract

We present a semi-Lagrangian method for the numerical resolution of Vlasov-type equations on multi-patch meshes. Following N. Crouseilles et al. [A parallel Vlasov solver based on local cubic spline interpolation on patches. Journal of Computational Physics (2009)], we employ a local cubic spline interpolation with Hermite boundary conditions between the patches. The derivative reconstruction is adapted to cope with non-uniform meshes as well as non-conforming situations. In the conforming case, there are no longer any constraints on the number of points for each patch; however, a small global system must now be solved. In that case, the local spline representations coincide with the corresponding global spline reconstruction. Alternatively, we can choose not to apply the global system and the derivatives can be approximated. The influence of the most distant points diminishes as the number of points per patch increases. For uniform per patch configurations, a study of the explicit and asymptotic behavior of this influence has been led. The method is validated using a two-dimensional guiding-center model with an O-point. All the numerical results are carried out in the Gyselalib++ library.