arXiv subjects

Hongying Huang

Publications and source records attributed to Hongying Huang.

5 recordsLinked to original sources

arXiv2026

A fully discrete LBRFD-IPDG method for linear fourth-order parabolic equations

We propose a fully discrete method for linear fourth-order parabolic equations with Dirichlet boundary conditions, combining an implicit LBRFD multistep scheme in time with a mixed interior penalty discontinuous Galerkin (IPDG) method in space. The temporal discretization employs equispaced linear barycentric rational interpolants and incorporates a startup procedure. To facilitate the spatial discretization, the original problem is reformulated through an auxiliary variable. For certain parameter pairs $(n,d)$, the LBRFD method is shown to be $A(\alpha)$-stable and to possess a wider stability angle than the corresponding BDF$p$ method of the same order. Stability and a priori error estimates are established via a $G$-energy technique and the discrete Gr\"onwall lemma. The theoretical analysis yields a total $L^2$ error estimate of order $h^{k-1}+\tau^p$, where $p=d$ if $n-d$ is even and $p=d+1$ if $n-d$ is odd. The reduced spatial convergence rate is attributed to boundary contributions on $\partial\Omega$. Despite this theoretical prediction, numerical experiments confirm the stability and demonstrate optimal convergence of order $h^{k+1}+\tau^p$.

math.NA↗

arXiv2026

A New $L2-1_{\sigma}$-Interior Penalty Method for Variable-Order Time-Fractional Subdiffusion Interface Problem with Curved Interface

This paper treats variable-order time-fractional subdiffusion with discontinuous coefficients across a curved interface using $L2\!-\!1_\sigma$ time stepping on graded meshes and a symmetric interior penalty FEM on body-fitted meshes. Stability and optimal a priori error estimates in a discrete-in-time $L^2$ norm are established, yielding second-order temporal accuracy. While analysis typically assumes $\alpha_n$ at $t_{n-\sigma_n}$ lies in the range of $\alpha(t)$ on $[t_{n-1},t_n]$ and $\alpha_n\le \alpha(t_{n-\alpha_n/2})$, experiments indicate the second inequality can be relaxed or omitted, enabling straightforward selection of $\alpha_n$ from many admissible values without solving a nonlinear equation. Numerical results verify temporal rates $\min\{2,r\delta\}$, spatial order $\min\{s,k+1\}$, and robustness to superconvergent points and interface geometry.

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arXiv2024

Determining superconvergence points for $L2-1_\sigma$ scheme of variable-exponent subdiffusion and error estimate

We develop a numerical scheme for subdiffusion of variable exponent by combining the $L2-1_\sigma$ temporal discretization with finite element spatial approximation. In existing works, determining the superconvergence points requires solving a nonlinear equation related to the variable exponent at each time step. This work relaxes the selection criterion of superconvergence points without affecting the numerical accuracy, which may reduce the cost of determining superconvergence points. To handle the initial singularity of the solution, we employ a graded temporal mesh. Then we prove the stability and error estimates with a convergence rate $O\left(N^{-\min\{r\delta,2\}}+h^{\mu}\right)$ for the $L2-1_\sigma$ scheme of variable-exponent subdiffusion. Numerical results are performed to substantiate the theoretical findings.

math.NA↗

arXiv2022

Cell-average based neural network method for high dimensional parabolic differential equations

In this paper, we introduce cell-average based neural network (CANN) method to solve high-dimensional parabolic partial differential equations. The method is based on the integral or weak formulation of partial differential equations. A feedforward network is considered to train the solution average of cells in neighboring time. Initial values and approximate solution at $t=Δt$ obtained by high order numerical method are taken as the inputs and outputs of network, respectively. We use supervised training combined with a simple backpropagation algorithm to train the network parameters. We find that the neural network has been trained to optimality for high-dimensional problems, the CFL condition is not strictly limited for CANN method and the trained network is used to solve the same problem with different initial values. For the high-dimensional parabolic equations, the convergence is observed and the errors are shown related to spatial mesh size but independent of time step size.

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arXiv2015

Third order Maximum-Principle-Satisfying Direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangle mesh

We develop 3rd order maximum-principle-satisfying direct discontinuous Galerkin methods [8, 9, 19, 21] for convection diffusion equations on unstructured triangular mesh. We carefully calculate the normal derivative numerical flux across element edges and prove that, with proper choice of parameter pair $(\beta_0,\beta_1)$ in the numerical flux, the quadratic polynomial solution satisfies strict maximum principle. The polynomial solution is bounded within the given range and third order accuracy is maintained. There is no geometric restriction on the meshes and obtuse triangles are allowed in the partition. A sequence of numerical examples are carried out to demonstrate the accuracy and capability of the maximum-principle-satisfying limiter.

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