Sparse-grid discontinuous Galerkin methods for the Vlasov–Poisson–Lenard–Bernstein model

• Published in: Journal of computational physics 2024-08, Vol.510, p.113053, Article 113053
• Main Authors: Schnake, Stefan, Kendrick, Coleman, Endeve, Eirik, Stoyanov, Miroslav, Hahn, Steven, Hauck, Cory D., Green, David L., Snyder, Phil, Canik, John
• Format: Article
• Language: English
• Subjects: Discontinuous Galerkin; Implicit-explicit; Kinetic equation; Lenard-Bernstein; MATHEMATICS AND COMPUTING; Sparse grids; Vlasov-Poisson
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2024.113053

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov–Poisson–Lenard–Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. We use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD.