arXiv subjects

Yulong Xing

Publications and source records attributed to Yulong Xing.

At least 19 recordsLinked to original sources

arXiv2026

Mesh-Based Filtering to Alleviate Time-Step Restrictions in Runge--Kutta Discontinuous Galerkin Methods in Spherical-polar Coordinates: Application to the Euler Equations

We propose a mesh-based filtering approach to alleviate the severe timestep restrictions arising in explicit Runge--Kutta discontinuous Galerkin (RKDG) methods formulated in spherical-polar coordinates. The filter enables stable evolution on the original logically Cartesian mesh while using larger time steps associated with an auxiliary merged mesh constructed to eliminate the extreme cell anisotropies produced by converging coordinate lines near coordinate singularities. The filter is implemented as a sequence of post-processing operations applied within an $s$-stage RK time integrator, making it straightforward to incorporate into existing structured-mesh DG frameworks. We analyze the filter in one spatial dimension and prove that the filtered RKDG method is equivalent to evolving the RKDG discretization on a nonuniform mesh obtained by merging selected elements of the underlying uniform mesh. This equivalence implies that the filtered method inherits the accuracy and stability properties of the corresponding RKDG discretization on the merged mesh. We apply the mesh-based filter to an existing RKDG method for the Euler equations in spherical-polar coordinates and demonstrate, through selected two- and three-dimensional examples, its effectiveness in accelerating simulations through significantly larger stable timesteps.

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arXiv2026

Hyperbolicity-Preserving Stochastic Galerkin Methods for Conservation Laws Based on Associative Truncated Products on Polynomial Spaces

Stochastic Galerkin discretizations of nonlinear hyperbolic conservation laws may lose hyperbolicity because the standard pseudospectral product is generally nonassociative, leading to non-commuting blocks in the flux Jacobian matrix. We develop a novel framework for constructing hyperbolicity-preserving stochastic Galerkin systems based on associative truncated products on polynomial spaces. In one stochastic dimension, we characterize associative truncated products through a single polynomial datum and identify examples with useful symmetry, positivity, and spectral properties, including collocation products and an associative symmetric product based on Gaussian quadrature nodes. We prove a consistency result showing that, under suitable projection-error assumptions, these products converge to the classical product as the polynomial degree grows. For systems with rational fluxes, we derive sufficient conditions under which the resulting stochastic Galerkin flux remains hyperbolic on the corresponding admissible set. Applications to the one-dimensional isothermal and compressible Euler equations show accurate statistical approximation and robust hyperbolicity preservation of the computed stochastic Galerkin states.

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arXiv2025

Realizability-Preserving Discontinuous Galerkin Method for Spectral Two-Moment Radiation Transport in Special Relativity

We present a realizability-preserving numerical method for solving a spectral two-moment model to simulate the transport of massless, neutral particles interacting with a steady background material moving with relativistic velocities. The model is obtained as the special relativistic limit of a four-momentum-conservative general relativistic two-moment model. Using a maximum-entropy closure, we solve for the Eulerian-frame energy and momentum. The proposed numerical method is designed to preserve moment realizability, which corresponds to moments defined by a nonnegative phase-space density. The realizability-preserving method is achieved with the following key components: (i) a discontinuous Galerkin (DG) phase-space discretization with specially constructed numerical fluxes in the spatial and energy dimensions; (ii) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (iii) a realizability-preserving conserved to primitive moment solver; (iv) a realizability-preserving implicit collision solver; and (v) a realizability-enforcing limiter. Component (iii) is necessitated by the closure procedure, which closes higher order moments nonlinearly in terms of primitive moments. The nonlinear conserved to primitive and the implicit collision solves are formulated as fixed-point problems, which are solved with custom iterative solvers designed to preserve the realizability of each iterate. With a series of numerical tests, we demonstrate the accuracy and robustness of this DG-IMEX method.

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arXiv2024

The Efficient Variable Time-stepping DLN Algorithms for the Allen-Cahn Model

We consider a family of variable time-stepping Dahlquist-Liniger-Nevanlinna (DLN) schemes, which is unconditional non-linear stable and second order accurate, for the Allen-Cahn equation. The finite element methods are used for the spatial discretization. For the non-linear term, we combine the DLN scheme with two efficient temporal algorithms: partially implicit modified algorithm and scalar auxiliary variable algorithm. For both approaches, we prove the unconditional, long-term stability of the model energy under any arbitrary time step sequence. Moreover, we provide rigorous error analysis for the partially implicit modified algorithm with variable time-stepping. Efficient time adaptive algorithms based on these schemes are also proposed. Several one- and two-dimensional numerical tests are presented to verify the properties of the proposed time adaptive DLN methods.

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arXiv2024

New Asymptotic Preserving, Hybrid Discontinuous Galerkin Methods the Radiation Transport Equation with Isotropic Scattering and Diffusive Scaling

Discontinuous Galerkin (DG) methods are widely adopted to discretize the radiation transport equation (RTE) with diffusive scalings. One of the most important advantages of the DG methods for RTE is their asymptotic preserving (AP) property, in the sense that they preserve the diffusive limits of the equation in the discrete setting, without requiring excessive refinement of the discretization. However, compared to finite element methods or finite volume methods, the employment of DG methods also increases the number of unknowns, which requires more memory and computational time to solve the problems. In this paper, when the spherical harmonic method is applied for the angular discretization, we perform an asymptotic analysis which shows that to retain the uniform convergence, it is only necessary to employ non-constant elements for the degree zero moment only in the DG spatial discretization. Based on this observation, we propose a heterogeneous DG method that employs polynomial spaces of different degrees for the degree zero and remaining moments respectively. To improve the convergence order, we further develop a spherical harmonics hybrid DG finite volume method, which preserves the AP property and convergence rate while tremendously reducing the number of unknowns. Numerical examples are provided to illustrate the effectiveness and accuracy of the proposed scheme.

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arXiv2024

The Runge--Kutta discontinuous Galerkin method with compact stencils for hyperbolic conservation laws

In this paper, we develop a new type of Runge--Kutta (RK) discontinuous Galerkin (DG) method for solving hyperbolic conservation laws. Compared with the original RKDG method, the new method features improved compactness and allows simple boundary treatment. The key idea is to hybridize two different spatial operators in an explicit RK scheme, utilizing local projected derivatives for inner RK stages and the usual DG spatial discretization for the final stage only. Limiters are applied only at the final stage for the control of spurious oscillations. We also explore the connections between our method and Lax--Wendroff DG schemes and ADER-DG schemes. Numerical examples are given to confirm that the new RKDG method is as accurate as the original RKDG method, while being more compact, for problems including two-dimensional Euler equations for compressible gas dynamics.

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arXiv2024

The Runge--Kutta discontinuous Galerkin method with stage-dependent polynomial spaces for hyperbolic conservation laws

In this paper, we present a novel class of high-order Runge--Kutta (RK) discontinuous Galerkin (DG) schemes for hyperbolic conservation laws. The new method extends beyond the traditional method of lines framework and utilizes stage-dependent polynomial spaces for the spatial discretization operators. To be more specific, two different DG operators, associated with $\mathcal{P}^k$ and $\mathcal{P}^{k-1}$ piecewise polynomial spaces, are used at different RK stages. The resulting method is referred to as the sdRKDG method. It features fewer floating-point operations and may achieve larger time step sizes. For problems without sonic points, we observe optimal convergence for all the sdRKDG schemes; and for problems with sonic points, we observe that a subset of the sdRKDG schemes remains optimal. We have also conducted von Neumann analysis for the stability and error of the sdRKDG schemes for the linear advection equation in one dimension. Numerical tests, for problems including two-dimensional Euler equations for gas dynamics, are provided to demonstrate the performance of the new method.

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arXiv2023

Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering

In highly diffusion regimes when the mean free path $\varepsilon$ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $\varepsilon^{-1}$ contribution, that leads to a nonuniform convergence for small $\varepsilon$. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a--priori estimates for the scaled spherical harmonic ($P_N$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some mild assumptions, its solutions converge uniformly in $\varepsilon$ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $\left(1+\mathcal{O}(\varepsilon)\right)h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used.

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arXiv2023

On a numerical artifact of solving shallow water equations with a discontinuous bottom: Analysis and a nontransonic fix

In this paper, we study a numerical artifact of solving the nonlinear shallow water equations with a discontinuous bottom topography. For various first-order schemes, the numerical solution of the momentum will form a spurious spike at the discontinuous points of the bottom, which should not exist in the exact solution. The height of the spike cannot be reduced even after the mesh is refined. For subsonic problems, this numerical artifact may cause the wrong convergence to a function far away from the exact solution. To explain the formation of the spurious spike, we perform a convergence analysis by proving a Lax--Wendroff type theorem. It is shown that the spurious spike is caused by the numerical viscosity in the computation of the water height at the discontinuous bottom. The height of the spike is proportional to the magnitude of the viscosity constant in the Lax--Friedrichs flux. Motivated by this conclusion, we propose a modified scheme by adopting the central flux at the bottom discontinuity in the equation of mass conservation, and show that this numerical artifact can be removed in many cases. For various numerical tests with nontransonic Riemann solutions, we observe that the modified scheme is able to retrieve the correct convergence.

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arXiv2022

High order asymptotic preserving well-balanced finite difference WENO schemes for all Mach full Euler equations with gravity

In this paper, we propose a high order semi-implicit well-balanced finite difference scheme for all Mach Euler equations with a gravitational source term. To obtain the asymptotic preserving property, we start from the conservative form of full compressible Euler equations and add the evolution equation of the perturbation of potential temperature. The resulting system is then split into a (non-stiff) nonlinear low dynamic material wave to be treated explicitly, and (stiff) fast acoustic and gravity waves to be treated implicitly. With the aid of explicit time evolution for the perturbation of potential temperature, we design a novel well-balanced finite difference WENO scheme for the conservative variables, which can be proven to be both asymptotic preserving and asymptotically accurate in the incompressible limit. Extensive numerical experiments were provided to validate these properties.

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arXiv2022

Energy conserving and well-balanced discontinuous Galerkin methods for the Euler-Poisson equations in spherical symmetry

This paper presents high-order Runge-Kutta (RK) discontinuous Galerkin methods for the Euler-Poisson equations in spherical symmetry. The scheme can preserve a general polytropic equilibrium state and achieve total energy conservation up to machine precision with carefully designed spatial and temporal discretizations. To achieve the well-balanced property, the numerical solutions are decomposed into equilibrium and fluctuation components which are treated differently in the source term approximation. One non-trivial challenge encountered in the procedure is the complexity of the equilibrium state, which is governed by the Lane-Emden equation. For total energy conservation, we present second- and third-order RK time discretization, where different source term approximations are introduced in each stage of the RK method to ensure the conservation of total energy. A carefully designed slope limiter for spherical symmetry is also introduced to eliminate oscillations near discontinuities while maintaining the well-balanced and total-energy-conserving properties. Extensive numerical examples -- including a toy model of stellar core-collapse with a phenomenological equation of state that results in core-bounce and shock formation -- are provided to demonstrate the desired properties of the proposed methods, including the well-balanced property, high-order accuracy, shock capturing capability, and total energy conservation.

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arXiv2022

Discontinuous Galerkin methods for stochastic Maxwell equations with multiplicative noise

In this paper we propose and analyze finite element discontinuous Galerkin methods for the one- and two-dimensional stochastic Maxwell equations with multiplicative noise. The discrete energy law of the semi-discrete DG methods were studied. Optimal error estimate of the semi-discrete method is obtained for the one-dimensional case, and the two-dimensional case on both rectangular meshes and triangular meshes under certain mesh assumptions. Strong Taylor 2.0 scheme is used as the temporal discretization. Both one- and two-dimensional numerical results are presented to validate the theoretical analysis results.

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arXiv2021

Multi-symplectic discontinuous Galerkin methods for the stochastic Maxwell equations with additive noise

One- and multi-dimensional stochastic Maxwell equations with additive noise are considered in this paper. It is known that such system can be written in the multi-symplectic structure, and the stochastic energy increases linearly in time. High order discontinuous Galerkin methods are designed for the stochastic Maxwell equations with additive noise, and we show that the proposed methods satisfy the discrete form of the stochastic energy linear growth property and preserve the multi-symplectic structure on the discrete level. Optimal error estimate of the semi-discrete DG method is also analyzed. The fully discrete methods are obtained by coupling with symplectic temporal discretizations. One- and two-dimensional numerical results are provided to demonstrate the performance of the proposed methods, and optimal error estimates and linear growth of the discrete energy can be observed for all cases.

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arXiv2021

Local discontinuous Galerkin methods for the abcd nonlinear Boussinesq system

Boussinesq type equations have been widely studied to model the surface water wave. In this paper, we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system, BBM-BBM system, Bona-Smith system etc. We propose local discontinuous Galerkin (LDG) methods, with carefully chosen numerical fluxes, to numerically solve this abcd Boussinesq system. The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a, b, c, d. Numerical experiments are shown to test the convergence rates, and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.

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arXiv2021

High order well-balanced asymptotic preserving finite difference WENO schemes for the shallow water equations in all Froude numbers

In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the "lake equations" for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.

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arXiv2021

Finite Element Approximations of a Class of Nonlinear Stochastic Wave Equation with Multiplicative Noise

Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite element method for a class of nonlinear stochastic wave equations, where the diffusion term is globally Lipschitz continuous while the drift term is only assumed to satisfy weaker conditions as in [11]. The novelties of this paper are threefold. First, the error estimates cannot not be directly obtained if the numerical scheme in primal form is used. The numerical scheme in mixed form is introduced and several Hölder continuity results of the strong solution are proved, which are used to establish the error estimates in both $L^2$ norm and energy norms. Second, two types of discretization of the nonlinear term are proposed to establish the $L^2$ stability and energy stability results of the discrete solutions. These two types of discretization and proper test functions are designed to overcome the challenges arising from the stochastic scaling in time issues and the nonlinear interaction. These stability results play key roles in proving the probability of the set on which the error estimates hold approaches one. Third, higher order moment stability results of the discrete solutions are proved based on an energy argument and the underlying energy decaying property of the method. Numerical experiments are also presented to show the stability results of the discrete solutions and the convergence rates in various norms.

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arXiv2021

High order finite difference Hermite WENO fixed-point fast sweeping method for static Hamilton-Jacobi equations

In this paper, we combine the nonlinear HWENO reconstruction in \cite{newhwenozq} and the fixed-point iteration with Gauss-Seidel fast sweeping strategy, to solve the static Hamilton-Jacobi equations in a novel HWENO framework recently developed in \cite{mehweno1}. The proposed HWENO frameworks enjoys several advantages. First, compared with the traditional HWENO framework, the proposed methods do not need to introduce additional auxiliary equations to update the derivatives of the unknown function $ϕ$. They are now computed from the current value of $ϕ$ and the previous spatial derivatives of $ϕ$. This approach saves the computational storage and CPU time, which greatly improves the computational efficiency of the traditional HWENO scheme. In addition, compared with the traditional WENO method, reconstruction stencil of the HWENO methods becomes more compact, their boundary treatment is simpler, and the numerical errors are smaller on the same mesh. Second, the fixed-point fast sweeping method is used to update the numerical approximation. It is an explicit method and does not involve the inverse operation of nonlinear Hamiltonian, therefore any Hamilton-Jacobi equations with complex Hamiltonian can be solved easily. It also resolves some known issues, including that the iterative number is very sensitive to the parameter $\varepsilon$ used in the nonlinear weights, as observed in previous studies. Finally, in order to further reduce the computational cost, a hybrid strategy is also presented. Extensive numerical experiments are performed on two-dimensional problems, which demonstrate the good performance of the proposed fixed-point fast sweeping HWENO methods.

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arXiv2021

High order asymptotic preserving Hermite WENO fast sweeping method for the steady-state $S_{N}$ transport equation

In this paper, we propose to combine the fifth order Hermite weighted essentially non-oscillatory (HWENO) scheme and fast sweeping method (FSM) for the solution of the steady-state $S_{N}$ transport equation in the finite volume framework. It is well-known that the $S_{N}$ transport equation asymptotically converges to a macroscopic diffusion equation in the limit of optically thick systems with small absorption and sources. Numerical methods which can preserve the asymptotic limit are referred to as asymptotic preserving methods. In the one-dimensional case, we provide the analysis to demonstrate the asymptotic preserving property of the high order finite volume HWENO method, by showing that its cell-edge and cell-average fluxes possess the thick diffusion limit. Numerical results in both one- and two- dimensions are presented to validate its asymptotic preserving property. A hybrid strategy to compute the nonlinear weights in the HWENO reconstruction is introduced to save computational cost. Extensive one- and two-dimensional numerical experiments are performed to verify the accuracy, asymptotic preserving property and positivity of the proposed HWENO FSM.

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