arXiv subjects

Nanyi Zheng

Publications and source records attributed to Nanyi Zheng.

8 recordsLinked to original sources

arXiv2026

A Mass, Momentum, and Energy Conserving Semi-Lagrangian Adaptive-Rank (SLAR) Method for the Vlasov-Poisson System

We propose a semi-Lagrangian adaptive-rank (SLAR) method that combines the large time-step capability of semi-Lagrangian schemes with the efficiency of adaptive-rank tensor representations while simultaneously enforcing local conservation laws for mass, momentum, and energy. The method builds on the high-dimensional SLAR framework introduced in our previous work and achieves high-order accuracy in both space and time. To address the loss of conservation in long-time simulations, we extend the implicit local macroscopic conservative (LoMaC) correction technique for the BGK equation to the high-dimensional Vlasov--Poisson (VP) system. The implicit macroscopic system is discretized using backward differentiation formulas and solved with a Jacobian-free Newton-Krylov method. This approach enables a consistent coupling with semi-Lagrangian methods which are capable of taking large time steps. A novel component of the proposed method is a unified adaptive-weight projection technique that eliminates the ad hoc parameter tuning required by previous LoMaC approaches. These weights capture problem-dependent velocity space structures and are constructed from the low-rank velocity bases of the solution. The local semi-Lagrangian method used in this work reconstructs the solution at the feet of the characteristics using efficient tensor contractions. To the best of our knowledge, this is the first successful implementation of an implicit LoMaC method for the VP system up to the 2D--2V setting. Numerical experiments on several classical benchmark problems demonstrate the accuracy and efficiency of the proposed method, as well as its ability to preserve conservation laws in VP simulations.

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arXiv2026

A Structure-preserving Adaptive-Rank Approach to the High-Dimensional Wigner-Poisson System

The Wigner-Poisson system is a deterministic phase-space model for quantum kinetic electron dynamics, but high-dimensional simulations are limited by the full 3D3V phase space and the nonlocal Wigner potential. We develop a structure-preserving, sampling-based adaptive-rank solver in hierarchical Tucker format for finite-$H$ regimes in which Wigner-Poisson solutions exhibit exploitable low-rank structure. The central difficulty is that adaptive compression can destroy the Fourier-Hermitian tensor symmetry required for a real inverse velocity transform and can break discrete global conservation laws. We address these issues with a Fourier-Hermitian-symmetry-aware sampling and mapping procedure and a global moment correction enforcing mass, momentum, and self-consistent total energy. Numerical tests for two-stream instability and strong Landau damping in 2D2V and 3D3V show roundoff-level conservation, preservation of the real-valued inverse transform, and approximately linear scaling with respect to the number of grid points per coordinate over the tested rank range. The results demonstrate that long-time 3D3V Wigner-Poisson simulations can be performed without assembling the full phase-space tensor.

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arXiv2025

A Semi-Lagrangian Adaptive Rank (SLAR) Method for High-Dimensional Vlasov Dynamics

We extend our previous work on a semi-Lagrangian adaptive rank (SLAR) integrator, in the finite difference framework for nonlinear Vlasov-Poisson systems, to the general high-order tensor setting. The proposed scheme retains the high-order accuracy of semi-Lagrangian methods, ensuring stability for large time steps and avoiding dimensional splitting errors. The primary contribution of this paper is the novel extension of the algorithm from the matrix to the high-dimensional tensor setting, which enables the simulation of Vlasov models in up to six dimensions. The key technical components include (1) a third-order high-dimensional polynomial reconstruction that scales as $O(d^2)$, providing a point-wise approximation of the solution at the foot of characteristics in a semi-Lagrangian scheme; (2) a recursive hierarchical adaptive cross approximation of high-order tensors in a hierarchical Tucker format, characterized by a tensor tree; (3) a low-complexity Poisson solver in the hierarchical Tucker format that leverages the FFT for efficiency. The computed adaptive rank kinetic solutions exhibit low-rank structures within branches of the tensor tree resulting in substantial computational savings in both storage and time. The resulting algorithm achieves a computational complexity of $O(d^4 N r^{3+\lceil\log_2d\rceil})$, where $N$ is the number of grid points per dimension, $d$ is the problem dimension, and $r$ is the maximum rank in the tensor tree, overcoming the curse of dimensionality. Through extensive numerical tests, we demonstrate the efficiency of the proposed algorithm and highlight its ability to capture complex solution structures while maintaining a computational complexity that scales linearly with $N$.

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arXiv2025

An Adaptive-rank Approach with Greedy Sampling for Multi-scale BGK Equations

In this paper, we propose a novel adaptive-rank method for simulating multi-scale BGK equations, based on a greedy sampling strategy. The method adaptively selects important rows and columns of the solution matrix and updates them using a local semi-Lagrangian solver. An adaptive cross approximation then reconstructs the full solution matrix. This extends our prior semi-Lagrangian adaptive-rank framework, developed for the Vlasov-Poisson system, to nonlinear collisional kinetic equations. Unlike step-and-truncate low-rank integrators, our greedy sampling approach avoids explicit low-rank decompositions of nonlinear terms, such as the local Maxwellian in the BGK operator. To ensure conservation, we introduce a locally macroscopic conservative correction that implicitly couples the kinetic and macroscopic systems, enforcing mass, momentum, and energy conservation. Through asymptotic analysis, we show that this correction preserves the full-grid scheme's asymptotic behavior, and that the proposed method is conditionally asymptotic-preserving in the low-rank setting. A key advantage of our approach is its use of a local semi-Lagrangian solver, which allows large time steps. This flexibility is retained in the macroscopic solver using high-order stiffly accurate diagonally implicit Runge-Kutta methods. The resulting nonlinear systems are solved efficiently using a Jacobian-free Newton-Krylov method, avoiding the need for preconditioning at modest CFL numbers. Each nonlinear iteration provides a self-consistent correction to a provisional kinetic solution, which serves as a dynamic closure for the macroscopic model. Numerical results demonstrate the method's accuracy in capturing shocks and its robustness across mixed-regime problems with wide-ranging Knudsen numbers.

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arXiv2025

A Sampling-Based Adaptive Rank Approach to the Wigner-Poisson System

We develop a mass-conserving, adaptive-rank solver for the 1D1V Wigner-Poisson system. Our work is motivated by applications to the study of the stopping power of $α$ particles at the National Ignition Facility (NIF). In this regime, electrons are in a warm dense state, requiring more than a standard kinetic model. They are hot enough to neglect Pauli exclusion, yet quantum enough to require accounting for uncertainty. The Wigner-Poisson system captures these effects but presents challenges due to its nonlocal nature. Based on a second-order Strang splitting method, we first design a full-rank solver with a structure-preserving Fourier update that ensures the intermediate solutions remain real-valued (up to machine precision), improving upon previous methods. Simulations demonstrate that the solutions exhibit a low rank structure for moderate to high dimensionless Planck constants ($H \ge 0.1$). This observed low rank structure motivates the development of an adaptive-rank solver, built on a Semi-Lagrangian adaptive-rank (SLAR) scheme for advection and an adaptive-rank, structure-preserving Fourier update for the Wigner integral terms, with a rigorous proof of structure-preserving property provided. Our solver achieves $O(N)$ complexity in both storage and computation time, while preserving mass and maintaining momentum accuracy up to the truncation error. The adaptive rank simulations are visually indistinguishable from the full-rank simulations in capturing solution structures. These results highlight the potential of adaptive rank methods for high-dimensional Wigner-Poisson simulations, paving the way toward fully kinetic studies of stopping power in warm dense plasmas.

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arXiv2024

A Semi-Lagrangian Adaptive-Rank (SLAR) Method for Linear Advection and Nonlinear Vlasov-Poisson System

High-order semi-Lagrangian methods for kinetic equations have been under rapid development in the past few decades. In this work, we propose a semi-Lagrangian adaptive rank (SLAR) integrator in the finite difference framework for linear advection and nonlinear Vlasov-Poisson systems without dimensional splitting. The proposed method leverages the semi-Lagrangian approach to allow for significantly larger time steps while also exploiting the low-rank structure of the solution. This is achieved through cross approximation of matrices, also referred to as CUR or pseudo-skeleton approximation, where representative columns and rows are selected using specific strategies. To maintain numerical stability and ensure local mass conservation, we apply singular value truncation and a mass-conservative projection following the cross approximation of the updated solution. The computational complexity of our method scales linearly with the mesh size $N$ per dimension, compared to the $\mathcal{O}(N^2)$ complexity of traditional full-rank methods per time step. The algorithm is extended to handle nonlinear Vlasov-Poisson systems using a Runge-Kutta exponential integrator. Moreover, we evolve the macroscopic conservation laws for charge densities implicitly, enabling the use of large time steps that align with the semi-Lagrangian solver. We also perform a mass-conservative correction to ensure that the adaptive rank solution preserves macroscopic charge density conservation. To validate the efficiency and effectiveness of our method, we conduct a series of benchmark tests on both linear advection and nonlinear Vlasov-Poisson systems. The propose algorithm will have the potential in overcoming the curse of dimensionality for beyond 2D high dimensional problems, which is the subject of our future work.

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arXiv2024

Non-splitting Eulerian-Lagrangian WENO schemes for two-dimensional nonlinear convection-diffusion equations

In this paper, we develop high-order, conservative, non-splitting Eulerian-Lagrangian (EL) Runge-Kutta (RK) finite volume (FV) weighted essentially non-oscillatory (WENO) schemes for convection-diffusion equations. The proposed EL-RK-FV-WENO scheme defines modified characteristic lines and evolves the solution along them, significantly relaxing the time-step constraint for the convection term. The main algorithm design challenge arises from the complexity of constructing accurate and robust reconstructions on dynamically varying Lagrangian meshes. This reconstruction process is needed for flux evaluations on time-dependent upstream quadrilaterals and time integrations along moving characteristics. To address this, we propose a strategy that utilizes a WENO reconstruction on a fixed Eulerian mesh for spatial reconstruction, and updates intermediate solutions on the Eulerian background mesh for implicit-explicit RK temporal integration. This strategy leverages efficient reconstruction and remapping algorithms to manage the complexities of polynomial reconstructions on time-dependent quadrilaterals, while ensuring local mass conservation. The proposed scheme ensures mass conservation due to the flux-form semi-discretization and the mass-conservative reconstruction on both background and upstream cells. Extensive numerical tests have been performed to verify the effectiveness of the proposed scheme.

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arXiv2022

Fourth-order conservative non-splitting semi-Lagrangian Hermite WENO schemes for kinetic and fluid simulations

We present fourth-order conservative non-splitting semi-Lagrangian (SL) Hermite essentially non-oscillatory (HWENO) schemes for linear transport equations with applications for nonlinear problems including the Vlasov-Poisson system, the guiding center Vlasov model, and the incompressible Euler equations in the vorticity-stream function formulation. The proposed SL HWENO schemes combine a weak formulation of the characteristic Galerkin method with two newly constructed HWENO reconstruction methods. Fourth-order accuracy is accomplished in both space and time under a non-splitting setting. Mass conservation naturally holds due to the weak formulation of the characteristic Galerkin method and the design of the HWENO reconstructions. We apply a positive-preserving limiter to maintain the positivity of numerical solutions when needed. Although the proposed SL framework allows us to take large time steps for improving computational efficiency, it also brings challenges to the spatial reconstruction technique; we construct two kind of novel HWENO reconstructions to fit the need for the proposed SL framework. Abundant benchmark tests are performed to verify the effectiveness of the proposed SL HWENO schemes.

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