arXiv subjects

Xiaofeng Cai

Publications and source records attributed to Xiaofeng Cai.

17 recordsLinked to original sources

arXiv2025

A Third-order Conservative Semi-Lagrangian Discontinuous Galerkin Scheme For the Transport Equation on Curvilinear Unstructured Meshes

We develop a third-order conservative semi-Lagrangian discontinuous Galerkin (SLDG) scheme for solving linear transport equations on curvilinear unstructured triangular meshes, tailored for complex geometries. To ensure third-order spatial accuracy while strictly preserving mass, we develop a high-order conservative intersection-based remapping algorithm for curvilinear unstructured meshes, which enables accurate and conservative data transfer between distinct curvilinear meshes. Incorporating this algorithm, we construct a non-splitting high-order SLDG method equipped with weighted essentially non-oscillatory and positivity-preserving limiters to effectively suppress numerical oscillations and maintain solution positivity. For the linear problem, the semi-Lagrangian update enables large time stepping, yielding an explicit and efficient implementation. Rigorous numerical analysis confirms that our scheme achieves third-order accuracy in both space and time, as validated by consistent error analysis in terms of $L^1$ and $L^2$-norms. Numerical benchmarks, including rigid body rotation and swirling deformation flows with smooth and discontinuous initial conditions, validate the scheme's accuracy, stability, and robustness.

math.NA↗

arXiv2025

Asymptotic-preserving conservative semi-Lagrangian discontinuous Galerkin schemes for the Vlasov-Poisson system in the quasi-neutral limit

We discretize the Vlasov-Poisson system using conservative semi-Lagrangian (CSL) discontinuous Galerkin (DG) schemes that are asymptotic preserving (AP) in the quasi-neutral limit. The proposed method (CSLDG) relies on two key ingredients: the CSLDG discretization and a reformulated Poisson equation (RPE). The use of the CSL formulation ensures local mass conservation and circumvents the Courant-Friedrichs-Lewy condition, while the DG method provides high-order accuracy for capturing fine-scale phase space structures of the distribution function. The RPE is derived by the Poisson equation coupled with moments of the Vlasov equation. The synergy between the CSLDG and RPE components makes it possible to obtain reliable numerical solutions, even when the spatial and temporal resolution might not fully resolve the Debye length. We rigorously prove that the proposed method is asymptotically stable, consistent and satisfies AP properties. Moreover, its efficiency is maintained across non-quasi-neutral and quasi-neutral regimes. These properties of our approach are essential for accurate and robust numerical simulation of complex electrostatic plasmas. Several numerical experiments verify the accuracy, stability and efficiency of the proposed CSLDG schemes.

math.NA↗

arXiv2025

Asymptotic-preserving semi-Lagrangian discontinuous Galerkin schemes for the Boltzmann equation

In this work, we present an asymptotic-preserving semi-Lagrangian discontinuous Galerkin scheme for the Boltzmann equation that effectively handles multi-scale transport phenomena. The main challenge lies in designing appropriate moments update for penalization within the semi-Lagrangian framework. Inspired by [M. Ding, J. M. Qiu, and R. Shu, Multiscale Model. Simul. 21 (2023), no. 1, 143--167], the key ingredient is utilizing the Shu-Osher form of the scheme in the implicit-explicit Runge-Kutta (IMEX-RK) setting, which enables us to capture the correct limiting system by constructing an appropriate moments update procedure. Our theoretical analysis establishes accuracy order conditions for both the IMEX-RK time integration and the new moments update step. We also employ hypocoercivity techniques to establish stability for the linearized model. Numerical experiments for various test problems validate our proposed scheme's accuracy, asymptotic-preserving property, and robustness in various regimes, which demonstrates its effectiveness for multi-scale kinetic simulations.

math.NA↗

arXiv2025

A high-order energy-conserving semi-Lagrangian discontinuous Galerkin method for the Vlasov-Ampere system

In this paper, we propose a high-order energy-conserving semi-Lagrangian discontinuous Galerkin(ECSLDG) method for the Vlasov-Ampere system. The method employs a semi-Lagrangian discontinuous Galerkin scheme for spatial discretization of the Vlasov equation, achieving high-order accuracy while removing the Courant-Friedrichs-Lewy (CFL) constraint. To ensure energy conservation and eliminate the need to resolve the plasma period, we adopt an energy-conserving time discretization introduced by Liu et al. [J. Comput. Phys., 492 (2023), 112412]. Temporal accuracy is further enhanced through a high-order operator splitting strategy, yielding a method that is high-order accurate in both space and time. The resulting ECSLDG scheme is unconditionally stable and conserves both mass and energy at the fully discrete level, regardless of spatial or temporal resolution. Numerical experiments demonstrate the accuracy, stability, and conservation properties of the proposed method. In particular, the method achieves more accurate enforcement of Gauss's law and improved numerical fidelity over low-order schemes, especially when using a large CFL number.

math.NA↗

arXiv2025

The positivity-preserving high-order semi-Lagrangian spectral volume method for Vlasov-Poisson equations

In this paper, a novel high order semi-Lagrangian (SL) spectral volume (SV) method is proposed and studied for nonlinear Vlasov-Poisson (VP) simulations via operator splitting. The proposed algorithm combines both advantages of semi-Lagrangian and spectral volume approaches, exhibiting strong stability, robustness under large time steps, arbitrary high-order accuracy in space, local mass conservation, and positivity preservation. Numerical study of the SLSV method applied to the one-dimensional and two-dimensional transport equations, the Vlasov-Poisson system, the classical benchmark problems including Landau damping and two-stream instabilities is conducted, confirming the effectiveness, accuracy, and robustness of our algorithm in addressing complex nonlinear phenomena.

math.NA↗

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.

math.NA↗

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.

math.NA↗

arXiv2020

High Order Semi-Lagrangian Discontinuous Galerkin Method Coupled with Runge-Kutta Exponential Integrators for Nonlinear Vlasov Dynamics

In this paper, we propose a semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinear Vlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators (EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlinear transport setting, the RKEI can be used to decompose the evolution of the nonlinear transport into a composition of a sequence of linearized dynamics. The resulting linearized transport equations can be solved by the semi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al. (JSC, 2017). The proposed method can achieve high order spatial accuracy via the SLDG framework, and high order temporal accuracy via the RK EI. Due to the SL nature, the proposed SLDG-RKEI method is not subject to the CFL condition, thus they have the potential in using larger time-stepping sizes than those in the Eulerian approach. Inheriting advantages from the SLDG method, the proposed SLDG-RKEI schemes are mass conservative, positivity-preserving, have no dimensional splitting error, perform well in resolving complex solution structures, and can be evolved with adaptive time-stepping sizes. We show the performance of the SLDG-RKEI algorithm by classical test problems for the nonlinear Vlasov-Poisson system, as well as the Guiding center Vlasov model. Though that it is not our focus of this paper to explore the SLDG-RKEI scheme for nonlinear hyperbolic conservation laws that develop shocks, we show some preliminary results on schemes' performance on the Burgers' equation.

math.NA↗

arXiv2020

Semi-Lagrangian implicit Bhatnagar-Gross-Krook collision model for the finite-volume discrete Boltzmann method

A new implicit BGK collision model using a semi-Lagrangian approach is proposed in this paper. Unlike existing models, in which the implicit BGK collision is resolved either by a temporal extrapolation or by a variable transformation, the new model removes the implicitness by tracing the particle distribution functions (PDFs) back in time along their characteristic paths during the collision process. An interpolation scheme is needed to evaluate the PDFs at the traced-back locations. By using the first-order interpolation, the resulting model allows for the straightforward replacement of ${f_α}^{eq,n+1}$ by ${f_α}^{eq,n}$ no matter where it appears. After comparing the new model with the existing models under different numerical conditions (e.g. different flux schemes and time marching schemes) and using the new model to successfully modify the variable transformation technique, three conclusions can be drawn. First, the new model can improve the accuracy by almost an order of magnitude. Second, it can slightly reduce the computational cost. Therefore, the new scheme improves accuracy without extra cost. Finally, the new model can significantly improve the $Δt/τ$ limit compared to the temporal interpolation model while having the same $Δt/τ$ limit as the variable transformation approach. The new scheme with a second-order interpolation is also developed and tested; however, that technique displays no advantage over the simple first-order interpolation approach. Both numerical and theoretical analyses are also provided to explain why the new implicit scheme with simple first-order interpolation can outperform the same scheme with second-order interpolation, as well as the existing temporal extrapolation and variable transformation schemes.

physics.comp-ph↗

arXiv2020

An Eulerian-Lagrangian discontinuous Galerkin method for transport problems and its application to nonlinear dynamics

We propose a new Eulerian-Lagrangian (EL) discontinuous Galerkin (DG) method. The method is designed as a generalization of the semi-Lagrangian (SL) DG method for linear advection problems proposed in [J. Sci. Comput. 73: 514-542, 2017], which is formulated based on an adjoint problem and tracing upstream cells by tracking characteristics curves highly accurately. In the SLDG method, depending on the velocity field, upstream cells could be of arbitrary shape. Thus, a more sophisticated approximation to sides of the upstream cells is required to get high order approximation. For example, quadratic-curved (QC) quadrilaterals were proposed to approximate upstream cells for a third-order spatial accuracy in a swirling deformation example. In this paper, for linear advection problems, we propose a more general formulation, named the ELDG method. The scheme is formulated based on a {\em modified} adjoint problem for which the upstream cells are always quadrilaterals, which avoids the need to use QC quadrilaterals in the SLDG algorithm. The newly proposed ELDG method can be viewed as a new general framework, in which both the classical Eulerian Runge-Kutta DG formulation and the SL DG formulation can fit in. Numerical results on linear transport problems, as well as the nonlinear Vlasov and incompressible Euler dynamics using the exponential RK time integrators, are presented to demonstrate the effectiveness of the ELDG method.

math.NA↗

arXiv2020

A semi-Lagrangian discontinuous Galerkin (DG) -- local DG method for solving convection-diffusion equations

In this paper, we propose an efficient high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for solving linear convection-diffusion equations. The method generalizes our previous work on developing the SLDG method for transport equations (J. Sci. Comput. 73: 514-542, 2017), making it capable of handling additional diffusion and source terms. Within the DG framework, the solution is evolved along the characteristics; while the diffusion term is discretized by the local DG (LDG) method and integrated along characteristics by implicit Runge-Kutta methods together with source terms. The proposed method is named the `SLDG-LDG' method and enjoys many attractive features of the DG and SL methods. These include the uniformly high order accuracy (e.g. third order) in space and in time, compact, mass conservative, and stability under large time stepping size. An $L^2$ stability analysis is provided when the method is coupled with the first order backward Euler discretization. Effectiveness of the method are demonstrated by a group of numerical tests in one and two dimensions.

math.NA↗

arXiv2019

Comparison of semi-Lagrangian discontinuous Galerkin schemes for linear and nonlinear transport simulations

Transport problems arise across diverse fields of science and engineering. Semi-Lagrangian (SL) discontinuous Galerkin (DG) methods are a class of high order deterministic transport solvers that enjoy advantages of both SL approach and DG spatial discretization. In this paper, we review existing SLDG methods to date and compare numerical their performances. In particular, we make a comparison between the splitting and non-splitting SLDG methods for multi-dimensional transport simulations. Through extensive numerical results, we offer a practical guide for choosing optimal SLDG solvers for linear and nonlinear transport simulations.

math.NA↗

arXiv2018

A high order semi-Lagrangian discontinuous Galerkin method for the two-dimensional incompressible Euler equations and the guiding center Vlasov model without operator splitting

In this paper, we generalize a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for multi-dimensional linear transport equations without operator splitting developed in Cai et al. (J. Sci. Comput. 73: 514-542, 2017) to the 2D time dependent incompressible Euler equations in the vorticity-stream function formulation and the guiding center Vlasov model. We adopt a local DG method for Poisson's equation of these models. For tracing the characteristics, we adopt a high order characteristics tracing mechanism based on a prediction-correction technique. The SLDG with large time-stepping size might be subject to extreme distortion of upstream cells. To avoid this problem, we propose a novel adaptive time-stepping strategy by controlling the relative deviation of areas of upstream cells.

math.NA↗

arXiv2017

A high order conservative semi-Lagrangian discontinuous Galerkin method for two-dimensional transport simulations

In this paper, we develop a class of high order conservative semi-Lagrangian (SL) discontinuous Galerkin (DG) methods for solving multi-dimensional linear transport equations. The methods rely on a characteristic Galerkin weak formulation, leading to $L^2$ stable discretizations for linear problems. Unlike many existing SL methods, the high order accuracy and mass conservation of the proposed methods are realized in a non-splitting manner. Thus, the detrimental splitting error, which is known to significantly contaminate long term transport simulations, will be not incurred. One key ingredient in the scheme formulation, borrowed from CSLAM [Lauritzen, Nair, and Ullrich, 2010], is the use of Green's theorem which allows us to convert volume integrals into a set of line integrals. The resulting line integrals are much easier to approximate with high order accuracy, hence facilitating the implementation. Another novel ingredient is the construction of quadratic curves in approximating sides of upstream cell, leading to quadratic-curved quadrilateral upstream cells. Formal third order accuracy is obtained by such a construction. The desired positivity-preserving property is further attained by incorporating a high order bound-preserving filter. To assess the performance of the proposed methods, we test and compare the numerical schemes with a variety of configurations for solving several benchmark transport problems with both smooth and nonsmooth solutions. The efficiency and efficacy are numerically verified.

math.NA↗

arXiv2017

Finite volume HWENO schemes for nonconvex conservation laws

We illustrate that numerical solutions of high order finite volume Hermite weighted essentially non-oscillatory (HWENO) scheme for some nonconvex conservation laws perform poorly or converge to the entropy solution in a slow speed. The modified finite volume HWENO schemes based either on first order monotone schemes or a second order entropic projection following the work of Qiu and Shu [SIAM J. Sci. Comput., 31 (2008), 584-607] are proposed and compared for solving one-dimensional scalar problems. We extend the modified finite volume HWENO based on first order monotone schemes for one-dimensional systems and two-dimensional scalar conservation laws. Numerical tests for several representative examples will be reported.

math.NA↗

arXiv2017

A high order semi-Lagrangian discontinuous Galerkin method for Vlasov-Poisson simulations without operator splitting

In this paper, we develop a high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for nonlinear Vlasov-Poisson (VP) simulations without operator splitting. In particular, we combine two recently developed novel techniques: one is the high order non-splitting SLDG transport method [Cai, et al., J Sci Comput, 2017], and the other is the high order characteristics tracing technique proposed in [Qiu and Russo, J Sci Comput, 2017]. The proposed method with up to third order accuracy in both space and time is locally mass conservative, free of splitting error, positivity-preserving, stable and robust for large time stepping size. The SLDG VP solver is applied to classic benchmark test problems such as Landau damping and two-stream instabilities for VP simulations. Efficiency and effectiveness of the proposed scheme is extensively tested. Tremendous CPU savings are shown by comparisons between the proposed SL DG scheme and the classical Runge-Kutta DG method.

math.NA↗

arXiv2016

A conservative semi-Lagrangian HWENO method for the Vlasov equation

In this paper, we present a high order conservative semi-Lagrangian (SL) Hermite weighted essentially non-oscillatory (HWENO) method for the Vlasov equation based on dimensional splitting [Cheng and Knorr, Journal of Computational Physics, 22(1976)]. The major advantage of HWENO reconstruction, compared with the original WENO reconstruction, is compact. For the split one-dimensional equation, to ensure local mass conservation, we propose a high order SL HWENO scheme in a conservative flux-difference form, following the work in [J.-M. Qiu and A. Christlieb, Journal of Computational Physics, v229(2010)]. Besides performing dimensional splitting for the original 2D problem, we design a proper splitting for equations of derivatives to ensure local mass conservation of the proposed HWENO scheme. The proposed fifth order SL HWENO scheme with the Eulerian CFL condition has been tested to work well in capturing filamentation structures without introducing oscillations. We introduce WENO limiters to control oscillations when the time stepping size is larger than the Eulerian CFL restriction. We perform classical numerical tests on rigid body rotation problem, and demonstrate the performance of our scheme via the Landau damping and two-stream instabilities when solving the Vlasov-Poisson system.

math.NA↗