arXiv subjects

Zhengrong Xie

Publications and source records attributed to Zhengrong Xie.

5 recordsLinked to original sources

arXiv2026

Error Analysis of Time-Dependent Test Functions in the Semi-Lagrangian Discontinuous Finite Element Scheme Based on the Characteristic Galerkin Method

This paper studies the time-dependent test-function error in the characteristic Galerkin-type semi-Lagrangian discontinuous finite element (CSLDG) method caused by numerical integration errors of the characteristic ODE solver, and its effect on convergence. Unlike classical finite element methods and standard DG methods, the test functions in CSLDG are constructed by characteristic backtracking. As a result, ODE errors affect not only the upstream integration region but also enter directly into the discrete weak formulation through perturbations of the test function support. Neglecting mesh geometric errors, we introduce an ideal auxiliary solution and an auxiliary solution retaining the test-function error, and derive the corresponding error equation and recursive relations. First, based on a global time-dependent test-function error analysis, we show that the optimal $L^2$ convergence order of the $P^K$-CSLDG method is preserved provided that the ODE solver order satisfies $D\ge 2K+3+d$. We then propose a new analysis based on the propagation of local modal coefficient errors, which controls the error cell by cell through the actual modal update formulas and recovers the global $L^2$ estimate. This leads to an improved sufficient condition $D\ge K+1+\frac{d}{2}$. The results show that the required ODE accuracy depends strongly on the analytical route, and that the modal-coefficient-based analysis is both closer to the actual implementation and more consistent with numerical observations.

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arXiv2026

Numerical Analysis and Dimension Splitting for A Semi-Lagrangian Discontinuous Finite Element Scheme Based on the Characteristic Galerkin Method

A semi-Lagrangian discontinuous finite element scheme based on the characteristic Galerkin method (CSLDG) is investigated, which directly discretizes an integral invariant model derived from the coupling of the transport equation and its adjoint equation. First, the existence and stability of CSLDG are proven, along with the uniqueness of the numerical solution. Subsequently, in contrast to the commonly used interpolation-based dimensional splitting schemes (IBS) within the CSLDG framework, a separated-variable dimensional splitting approach based on the tensor product (SVS) is proposed and applied to the two-dimensional case. Numerical experiments show comparable accuracy between methods, but SVS demonstrates superior computational efficiency to IBS, especially on large-scale meshes.

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arXiv2026

Well-posedness and Regularity of the Integral Invariant Model from Linear Scalar Transport Equation

An integral invariant model derived from the coupling of the transport equation and its adjoint equation is investigated.Despite extensive research on the numerical implementation of this model,no studies have yet explored the well-posedness and regularity of the model itself.To address this gap,firstly, a comprehensive mathematical definition is formulated as a Cauchy initial value problem for the integral invariant model.This formulation preserves essential background information derived from relevant numerical algorithms.In the above definition,we directly evolve the time-dependent test function $ψ(\mathbf{x},t)$ through explicit construction rather than solving the adjoint equation,which enables reducing the required regularity of the test function $Ψ(\mathbf{x})$ from $C^1(Ω)$ to $L^2(Ω)$,contributing to stability proof. The challenge arising from the mismatch of integration domains on both sides of the model's equivalent form is overcome through the compact support property of test functions. For any arbitrary time instant $t^{*}\in[0,T]$,an abstract function $\mathcal{U}(λ)$ taking values in the Banach space $L^2(\mathbb{R}^d)$ is initially constructed on the entire space $\mathbb{R}^d$ via the Riesz representation theorem.Subsequently,this function is properly restricted to the time-dependent bounded domain $\widetildeΩ(t)$ through multiplication by the characteristic function. The existence of this model's solution in $L^{1}([0,T],L^2(\widetildeΩ(t)))$ is then rigorously established.Furthermore,by judiciously selecting test functions $Ψ$, the stability of the integral invariant model is proved, from which the uniqueness naturally follows.Finally, when the initial value $\widetilde{U}_0 \in L^{2}(\widetildeΩ(0))$,the temporal integrability of the model over $[0,T]$ can be enhanced to $L^{\infty}([0,T],L^2(\widetildeΩ(t)))$.

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arXiv2026

On the Spectral Properties of Van Leer and AUSM Flux-Vector Splitting Schemes

The flux-vector splitting scheme of Van Leer is a cornerstone of computational fluid dynamics, yet its original proof of the eigenvalue sign condition was presented in a condensed form. In this work, we provide a detailed and rigorous analysis of the eigenvalues of the Jacobian matrices associated with the Van Leer splitting for the one-dimensional Euler equations. By constructing the Sturm sequence of the discriminant, we prove that for the admissible parameter range $1 \le γ\le 3$, $|M|<1$, and $a>0$, the Jacobian $\partial F^+/\partial U$ has one zero eigenvalue and two positive real eigenvalues, confirming Van Leer's original claim. Furthermore, we extend our analysis to two variants of the original AUSM scheme (Advection Upstream Splitting Method) proposed by Liou and Steffen, considering both linear and second-order pressure splittings. For the linear pressure splitting we show that the eigenvalues are not all of the same sign, while for the second-order pressure splitting we prove that all coefficients of the characteristic equation are positive. Numerical experiments reported in the appendix confirm the non-negativity of the discriminant for the AUSM with the second-order pressure splitting, implying that its eigenvalues are real and positive.

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arXiv2024

Runge-Kutta Discontinuous Galerkin Method Based on Flux Vector Splitting with Constrained Optimization-based TVB(D)-minmod Limiter for Solving Hyperbolic Conservation Laws

The flux vector splitting (FVS) method has firstly been incorporated into the discontinuous Galerkin (DG) framework for reconstructing the numerical fluxes required for the spatial semi-discrete formulation, setting it apart from the conventional DG approaches that typically utilize the Lax-Friedrichs flux scheme or classical Riemann solvers. The control equations of hyperbolic conservation systems are initially reformulated into a flux-split form. Subsequently, a variational approach is applied to this flux-split form, from which a DG spatial semi-discrete scheme based on FVS is derived. In order to suppress numerical pseudo-oscillations, the smoothness measurement function IS from the WENO limiter is integrated into the TVB(D)-minmod limiter, constructing an optimization problem based on the smoothness factor constraint, thereby realizing a TVB(D)-minmod limiter applicable to arbitrary high-order polynomial approximation. Subsequently, drawing on the ``reconstructed polynomial and the original high-order scheme's L2 -error constraint'' from the literature [1] , combined with our smoothness factor constraint, a bi-objective optimization problem is formulated to enable the TVB(D)-minmod limiter to balance oscillation suppression and high precision. As for hyperbolic conservation systems, limiters are typically required to be used in conjunction with local characteristic decomposition. To transform polynomials from the physical space to the characteristic space, an interpolation-based characteristic transformation scheme has been proposed, and its equivalence with the original moment characteristic transformation has been demonstrated in one-dimensional scenarios. Finally, the concept of ``flux vector splitting based on Jacobian eigenvalue decomposition'' has been applied to the conservative linear scalar transport equations and the nonlinear Burgers' equation.

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