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Conservative Low-rank Approximations to nonlinear-vlasov Dynamics

Abstract: The Vlasov system is known as a fundamental model in plasma physics which describes the dynamics of dilute charged particles due to self-induced electrostatic forces. The main numerical challenges lie in the high dimensionality of the phase space, multi-scale feature, and the inherent conservation property of the solutions.

In this talk, I will introduce a novel conservative adaptive low-rank tensor method for the Vlasov system. The approach takes advantage of the fact that the differential operators in the Vlasov equation are tensor-friendly, based on which we propose to build up the low-rank solution basis dynamically and adaptively by exploring the intrinsic low-rank structure of Vlasov dynamics. We further develop a novel low-rank scheme with local mass, momentum, and energy conservation by considering the corresponding macroscopic equations. The mass and momentum conservation are achieved by a conservative low-rank truncation, while the energy conservation is achieved by replacing the energy component of the Vlasov solution by the one obtained from a conservative scheme for the macroscopic energy equation. The algorithm is extended to high-dimensional problems with the hierarchical Tucker tensor decomposition of Vlasov solutions, overcoming the curse of dimensionality. An extensive set of nonlinear Vlasov examples are performed to demonstrate the effectiveness and conservation property of the proposed conservative low-rank tensor approach.