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  1. In this paper, we propose a conservative low rank tensor method to approximate nonlinear Vlasov solutions. The low rank approach is based on our earlier work [W. Guo and J.-M. Qiu, A Low Rank Tensor Representation of Linear Transport and Nonlinear Vlasov Solutions and Their Associated Flow Maps, preprint, https://arxiv.org/abs/2106.08834, 2021]. It takes advantage of the fact that the differential operators in the Vlasov equation are tensor friendly, based on which we propose to dynamically and adaptively build up low rank solution basis by adding new basis functions from discretization of the differential equation, and removing basis from a singular value decomposition (SVD)-type truncation procedure. For the discretization, we adopt a high order finite difference spatial discretization together with a second order strong stability preserving multistep time discretization. While the SVD truncation will remove the redundancy in representing the high dimensional Vlasov solution, it will destroy the conservation properties of the associated full conservative scheme. In this paper, we develop a conservative truncation procedure with conservation of mass, momentum, and kinetic energy densities. The conservative truncation is achieved by an orthogonal projection onto a subspace spanned by 1, 𝑣, and 𝑣2 in the velocity space associated with a weighted inner product. Then the algorithm performs a weighted SVD truncation of the remainder, which involves a scaling, followed by the standard SVD truncation and rescaling back. The algorithm is further developed in high dimensions with hierarchical Tucker tensor decomposition of high dimensional Vlasov solutions, overcoming the curse of dimensionality. An extensive set of nonlinear Vlasov examples are performed to show the effectiveness and conservation property of proposed conservative low rank approach. Comparison is performed against the nonconservative low rank tensor approach on conservation history of mass, momentum, and energy. 
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    Free, publicly-accessible full text available February 28, 2025
  2. We develop a set of highly efficient and effective computational algorithms and simulation tools for fluid simulations on a network. The mathematical models are a set of hyperbolic conservation laws on edges of a network, as well as coupling conditions on junctions of a network. For example, the shallow water system, together with flux balance and continuity conditions at river intersections, model water flows on a river network. The computationally accurate and robust discontinuous Galerkin methods, coupled with explicit strong stability preserving Runge-Kutta methods, are implemented for simulations on network edges. Meanwhile, linear and nonlinear scalable Riemann solvers are being developed and implemented at network vertices. These network simulations result in tools that are added to the existing PETSc and DMNetwork software libraries for the scientific community in general. Simulation results of a shallow water system on a Mississippi river network with over one billion network variables are performed on an extreme-scale computer using up to 8,192 processor with an optimal parallel efficiency. Further potential applications include traffic flow simulations on a highway network and blood flow simulations on a arterial network, among many others. 
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