An implicit-in-time DPG formulation of the 1D1V Vlasov-Poisson equations

Journal Article · 23 November 2023 · Computers and Mathematics with Applications (Oxford)

DOI: https://doi.org/10.1016/j.camwa.2023.11.014 · OSTI ID:2311509

Roberts, Nathan V.  ^[1]; Miller, Sean T. ^[2]; Bond, Stephen D.  ^[1]; Cyr, Eric C. ^[1]

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1. Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
2. Advanced Micro Devices, Inc., Santa Clara, CA (United States)

Efficient solution of the Vlasov equation, which can be up to six-dimensional, is key to the simulation of many difficult problems in plasma physics. The discontinuous Petrov-Galerkin (DPG) finite element methodology provides a framework for the development of stable (in the sense of Ladyzhenskaya–Babuška–Brezzi conditions) finite element formulations, with built-in mechanisms for adaptivity. While DPG has been studied extensively in the context of steady-state problems and to a lesser extent with space-time discretizations of transient problems, relatively little attention has been paid to time-marching approaches. In the present work, we study a first application of time-marching DPG to the Vlasov equation, using backward Euler for a Vlasov-Poisson discretization. We demonstrate adaptive mesh refinement for two problems: the two-stream instability problem, and a cold diode problem. Furthermore, we believe the present work is novel both in its application of unstructured adaptive mesh refinement (as opposed to block-structured adaptivity, which has been studied previously) in the context of Vlasov-Poisson, as well as in its application of DPG to the Vlasov-Poisson system. We also discuss extensive additions to the Camellia library in support of both the present formulation as well as extensions to higher dimensions, Maxwell equations, and space-time formulations.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA); USDOE Laboratory Directed Research and Development (LDRD) Program
• Grant/Contract Number: NA0003525
• OSTI ID: 2311509
• Alternate ID(s): OSTI ID: 2369119
• Report Number(s): SAND--2023-13975J
• Journal Information: Computers and Mathematics with Applications (Oxford), Journal Name: Computers and Mathematics with Applications (Oxford) Vol. 154; ISSN 0898-1221
• Publisher: ElsevierCopyright Statement
• Country of Publication: United States
• Language: English