arXiv subjects

Weizhu Bao

Publications and source records attributed to Weizhu Bao.

At least 19 recordsLinked to original sources

arXiv2026

A spectral-element method for computing many eigenvalues and their asymptotics of the Schrödinger operator with Robin boundary condition

We propose spectral and spectral-element methods for accurately computing many eigenvalues of the Schrödinger operator with Robin boundary condition on simple and complex geometries, respectively. Due to their spectral-type accuracy in approximating those eigenfunctions corresponding to high-index eigenvalues, which are usually highly oscillatory, the proposed approaches have excellent resolution in computing thousands of eigenvalues accurately and efficiently with the resolution property that the number of eigenvalues with reasonable accuracy is proportional to the number of degrees of freedom. Using a standard HPC node with 128 GB of memory, we can obtain numerically more than 5,000 reliable eigenvalues with relative errors below $10^{-8}$ for different complex domains in two dimensions (2D). Based on these computed eigenvalues, we first confirm some theoretical results on the Robin-to-Neumann (RtN) gaps of the Laplacian operator in 2D, which were recently studied by Rudnick \textit{et al.} [\textit{Comm. Math. Phys.} 388 (2021)]. Then we systematically study the RtN gaps of the Schrödinger operator and their convergence rates. Based on our extensive numerical results, we formulate a unified conjecture on the cumulative averages of the RtN gaps of the Schrödinger operator.

math.NA↗

arXiv2026

A discrete duality finite volume method with harmonic average for semiconductor drift-diffusion equations

The stationary drift-diffusion model is widely used to model charge transport in semiconductor devices. Classical methods, such as the finite volume Scharfetter--Gummel (FVSG) method, perform well on high-quality Delaunay meshes but struggle on irregular or distorted meshes due to their reliance on Voronoi diagrams. To overcome this mesh limitation, this article introduces a new approach that integrates harmonic average stabilization into the discrete duality finite volume method (DDFV-HA). To validate our scheme, we compare DDFV-HA and FVSG for semiconductor simulations on both high- and low-quality meshes. Experiments show that DDFV-HA matches FVSG on high-quality meshes and is more reliable and accurate on low-quality meshes. Applying DDFV-HA to a real-world thyristor further confirms that it is well-suited for semiconductor simulations in complex, irregular domains where high-quality meshes are not easy to generate.

physics.comp-ph↗

arXiv2026

Optimal convergence rates of the Klein-Gordon-Schrödinger system in the nonrelativistic limit

In this paper, we study the Klein-Gordon-Schrödinger system in the nonrelativistic regime $ε\to 0$, where $ε$ is proportional to the inverse of the speed of light. We show that the Klein-Gordon-Schrödinger system converges to a system of decoupled linear Schrödinger equations over a long time interval of order $ε^{-1}$ with error estimates of the form $(1+t)ε^2$; in particular, the error estimate for the Schrödinger component is uniform in time of the form $ε^{2}$ The specific forms of the error estimates coincide with the numerical results shown by Bao et a.l., and the $O(ε^{2})$ convergence rates coincide with the order of initial error, and thus are optimal.

math.AP↗

arXiv2026

An energy-stable parametric finite element method for the Willmore flow in three dimensions

This work develops novel energy-stable parametric finite element methods (ES-PFEM) for the Willmore flow and curvature-dependent geometric gradient flows of surfaces in three dimensions. The key to achieving the energy stability lies in the use of two novel geometric identities: (i) a reformulated variational form of the normal velocity field, and (ii) incorporation of the temporal evolution of the mean curvature into the governing equations. These identities enable the derivation of a new variational formulation. By using the parametric finite element method, an implicit fully discrete scheme is subsequently developed, which maintains the energy dissipative property at the fully discrete level. Based on the ES-PFEM, comprehensive insights into the design of ES-PFEM for general curvature-dependent geometric gradient flows and a new understanding of mesh quality improvement in PFEM are provided. In particular, we develop the first PFEM for the Gauss curvature flow of surfaces. Furthermore, a tangential velocity control methodology is applied to improve the mesh quality and enhance the robustness of the proposed numerical method. Extensive numerical experiments confirm that the proposed method preserves energy dissipation properties and maintain good mesh quality in the surface evolution under the Willmore flow.

math.NA↗

arXiv2026

Optimal error bounds on the exponential wave integrator for nonlinear Schrödinger equations with highly singular potential

We establish error estimates of the first-order exponential wave integrator (EWI) for the nonlinear Schrödinger equation (NLSE) with a highly singular potential in $\mathbb{R}^d$ with $1\leq d \leq 3$. Our results deal with singular potentials in $L^p_\text{loc}(\mathbb{R}^d)$ with $p>\frac{d}{2}$ and $p\geq 1$, which is (almost) the weakest regularity of the potential required by the well-posedness of the NLSE. First, for $L^p_\text{loc}$-potentials with $p>2$, we establish an optimal first-order $L^2$-norm convergence for the EWI, with the convergence order slightly reduced to $1^-$ when $p=2$. To the best of our knowledge, the optimal first-order convergence for the three-dimensional $L^2$-potential is for the first time in the literature. The optimality of such an error bound is two-fold: (i) the first-order $L^2$-norm convergence is optimal for the EWI (and its higher-order versions) under the given $L^2$-regularity assumption on the potential, and (ii) to achieve the first-order $L^2$-norm convergence for the EWI, such an assumption is optimally weak. For more singular potentials in $L^p_\text{loc}(\mathbb{R}^d)$ with $\frac{d}{2} < p < 2$ and $p\geq 1$, we prove that the $L^2$-norm convergence is (almost) of $(1-α)$-order when $d=1,2$, and of $(1-\frac{3}{2}α)$-order when $d=3$, where $α:=d(1/p - 1/2)$ when $d =1,2,3$, $p>1$ and $α:=\frac{1}{2}^+$ when $d=1$, $p=1$. Notably, this result pushes the error estimate to the threshold regularity of the potential that matches the threshold regularity for the well-posedness of the NLSE, which is also for the first time. Two main ingredients are adopted in the proof: (i) the use of discrete space-time Lebesgue spaces together with discrete Strichartz estimates to establish the stability of the numerical scheme, and (ii) the use of normal form transformation and frequency decompositions to obtain optimal error bounds.

math.NA↗

arXiv2026

A structure-preserving parametric approximation for anisotropic geometric flows via an $α$-surface energy matrix

We propose a structure-preserving parametric approximation for geometric flows with general anisotropic effects. By introducing a hyperparameter $α$, we construct a unified surface energy matrix $\hat{\boldsymbol{G}}_k^α(θ)$ that encompasses all existing formulations of surface energy matrices, and apply it to anisotropic curvature flow. We prove that $α=-1$ is the unique choice achieving optimal energy stability under the necessary and sufficient condition $3\hatγ(θ)\geq\hatγ(θ-π)$, while all other $α\neq-1$ require strictly stronger conditions. The framework extends naturally to general anisotropic geometric flows through a unified velocity discretization that ensures energy stability. Numerical experiments validate the theoretical optimality of $α=-1$ and demonstrate the effectiveness and robustness.

math.NA↗

arXiv2026

Error estimates of an exponential wave integrator for the nonlinear Schrödinger equation with singular potential

We analyze a first-order exponential wave integrator (EWI) for the nonlinear Schrödinger equation (NLSE) with a singular potential that is locally in $L^2$, which might be locally unbounded. A typical example is the inverse power potential such as the Coulomb potential, which is the most fundamental potential in quantum physics and chemistry. We prove that, under the assumption of $L^2$-potential and $H^2$-initial data, the $L^2$-norm convergence of the EWI is, roughly, first-order in one dimension (1D) and two dimensions (2D), and $\frac{3}{4}$-order in three dimensions (3D). In addition, under a stronger integrability assumption of $L^p$-potential for some $p>2$ in 3D, the $L^2$-norm convergence increases to almost ${\frac{3}{4}} + 3(\frac{1}{2} - \frac{1}{p})$ order if $p \leq \frac{12}{5}$ and becomes first-order if $p > \frac{12}{5}$. In particular, our results show, to the best of our knowledge for the first time, that first-order $L^2$-norm convergence can be achieved when solving the NLSE with the Coulomb potential in 3D. The key advancements are the use of discrete (in time) Strichartz estimates, which allow us to handle the loss of integrability due to the singular potential that does not belong to $L^\infty$, and the more favorable local truncation error of the EWI, which requires no (spatial) smoothness of the potential. Extensive numerical results in 1D, 2D, and 3D are reported to confirm our error estimates and to show the sharpness of our assumptions on the regularity of the singular potentials.

math.NA↗

arXiv2026

A uniformly accurate multiscale time integrator for the nonlinear Klein-Gordon equation in the nonrelativistic regime via simplified transmission conditions

We propose a new and simplified multiscale time integrator Fourier pseudospectral (MTI-FP) method for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter epsilon in (0,1] inversely proportional to the speed of light, and establish its uniform first-order accuracy in time in the nonrelativistic regime, i.e. 0 < epsilon << 1. In this regime, the solution of the NKGE is highly oscillatory in time with O(epsilon^2)-wavelength, which brings significant difficulties in designing uniformly accurate numerical methods. The MTI-FP is based on (i) a multiscale decomposition by frequency of the NKGE in each time interval with simplified transmission conditions, and (ii) an exponential wave integrator for temporal discretization and a Fourier pseudospectral method for spatial discretization. By adapting the energy method and the mathematical induction, we obtain two error bounds in H1-norm at O(h^{m0}+tau^2/epsilon^2) and O(h^{m0}+tau+epsilon^2) with mesh size h, time step tau and m0 an integer dependent on the regularity of the solution, which immediately implies a uniformly accurate error bound O(h^{m0}+tau) with respect to epsilon in (0,1]. In addition, by adopting a linear interpolation of the micro variables with the multiscale decomposition in each time interval, we obtain a uniformly accurate numerical solution for any time t larger than zero. Thus the proposed MTI-FP method has a super resolution property in time in terms of the Shannon sampling theory, i.e. accurate numerical solutions can be obtained even when the time step is much bigger than the O(epsilon^2)-wavelength. Extensive numerical results are reported to confirm our error bounds and demonstrate their super resolution in time. Finally the proposed MTI-FP method is applied to study numerically convergence rates of the NKGE to its different limiting models in the nonrelativistic regime.

math.NA↗

arXiv2025

Optimal error bounds on an exponential wave integrator Fourier spectral method for the logarithmic Schrödinger equation

We prove a nearly optimal error bound on the exponential wave integrator Fourier spectral (EWI-FS) method for the logarithmic Schrödinger equation (LogSE) under the assumption of $H^2$-solution, which is theoretically guaranteed. Subject to a CFL-type time step size restriction $τ|\ln τ| \lesssim h^2/|\ln h|$ for obtaining the stability of the numerical scheme affected by the singularity of the logarithmic nonlinearity, an $L^2$-norm error bound of order $O(τ|\ln τ|^2 + h^2 |\ln h|)$ is established, where $τ$ is the time step size and $h$ is the mesh size. Compared to the error estimates of the LogSE in the literature, our error bound either greatly improves the convergence rate under the same regularity assumptions or significantly weakens the regularity requirement to obtain the same convergence rate. Moreover, our result can be directly applied to the LogSE with low regularity $L^\infty$-potential, which is not allowed in the existing error estimates. Two main ingredients are adopted in the proof: (i) an $H^2$-conditional $L^2$-stability estimate, which is established using the energy method to avoid singularity of the logarithmic nonlinearity, and (ii) mathematical induction with inverse inequalities to control the $H^2$-norm of the numerical solution. Numerical results are reported to confirm our error estimates and demonstrate the necessity of the time step size restriction imposed. We also apply the EWI-FS method to investigate soliton collisions in one dimension and vortex dipole dynamics in two dimensions.

math.NA↗

arXiv2025

A normalized gradient flow method for computing ground states of spin-2 Bose-Einstein condensates

We propose and analyze an efficient and accurate numerical method for computing ground states of spin-2 Bose-Einstein condensates (BECs) by using the normalized gradient flow (NGF). In order to successfully extend the NGF to spin-2 BECs which has five components in the vector wave function but with only two physical constraints on total mass conservation and magnetization conservation, two important techniques are introduced for designing the proposed numerical method. The first one is to systematically investigate the ground state structure and property of spin-2 BECs within a spatially uniform system, which can be used on how to properly choose initial data in the NGF for computing ground states of spin-2 BECs. The second one is to introduce three additional projection conditions based on the relations between the chemical potentials, together with the two existing physical constraints, such that the five projection parameters used in the projection step of the NGF can be uniquely determined. Then a backward-forward Euler finite difference method is adapted to discretize the NGF. We prove rigorously that there exists a unique solution of the nonlinear system for determining the five projection parameters in the full discretization of the NGF under a mild condition on the time step size. Extensive numerical results on ground states of spin-2 BECs with different types of phases and under different potentials are reported to show the efficiency and accuracy of the proposed numerical method and to demonstrate several interesting physical phenomena on ground states of spin-2 BECs.

math.NA↗

arXiv2024

Computing ground states of Bose-Einstein condensation by normalized deep neural network

We propose a normalized deep neural network (norm-DNN) for computing ground states of Bose-Einstein condensation (BEC) via the minimization of the Gross-Pitaevskii energy functional under unitary mass normalization. Compared with the traditional deep neural network for solving partial differential equations, two additional layers are added in training our norm-DNN for solving this kind of unitary constraint minimization problems: (i) a normalization layer is introduced to enforce the unitary mass normalization, and (ii) a shift layer is added to guide the training to non-negative ground state. The proposed norm-DNN gives rise to an efficient unsupervised approach for learning ground states of BEC. Systematical investigations are first carried out through extensive numerical experiments for computing ground states of BEC in one dimension. Extensions to high dimensions and multi-component are then studied in details. The results demonstrate the effectiveness and efficiency of norm-DNN for learning ground states of BEC. Finally, we extend the norm-DNN for computing the first excited states of BEC and discuss parameter generalization issues as well as compare with some existing machine learning methods for computing ground states of BEC in the literature.

cond-mat.quant-gas↗

arXiv2024

A structure-preserving parametric finite element method for solid-state dewetting on curved substrates

We consider a two-dimensional sharp-interface model for solid-state dewetting of thin films with anisotropic surface energies on curved substrates, where the film/vapor interface and substrate surface are represented by an evolving and a static curve, respectively. The model is governed by the anisotropic surface diffusion for the evolving curve, with appropriate boundary conditions at the contact points where the two curves meet. The continuum model obeys an energy decay law and preserves the enclosed area between the two curves. We introduce an arclength parameterization for the substrate curve, which plays a crucial role in a structure-preserving approximation as it straightens the curved substrate and tracks length changes between contact points. Based on this insight, we introduce a symmetrized weak formulation which leads to an unconditional energy stable parametric approximation in terms of the discrete energy. We also provide an error estimate of the enclosed area, which depends on the substrate profile and can be zero in the case of a flat substrate. Furthermore, we introduce a correction to the discrete normals to enable an exact area preservation for general curved substrates. The resulting nonlinear system is efficiently solved using a hybrid iterative algorithm which combines both Picard and Newton's methods. Numerical results are presented to show the robustness and good properties of the introduced method for simulating solid-state dewetting on various curved substrates.

math.NA↗

arXiv2024

Dynamics of Small Solid Particles on Substrates of Arbitrary Topography

We study the dynamics of a small solid particle arising from the dewetting of a thin film on a curved substrate driven by capillarity, where mass transport is controlled by surface diffusion. We consider the case when the size of the deposited particle is much smaller than the local radius of curvature of the substrate surface. The application of the Onsager variational principle leads to a reduced-order model for the dynamic behaviour of particles on arbitrarily curved substrates. We demonstrate that particles move toward region of the substrate surface with lower mean curvature with a determined velocity. In particular, the velocity is proportional to the substrate curvature gradient and inversely proportional to the size of the particle, with a coefficient that depends on material properties that include the surface energy, surface diffusivity, density, and Young's (wetting) angle. The reduced model is validated by comparing with numerical results for the full, sharp-interface model in both two and three dimensions.

cond-mat.mtrl-sci↗

arXiv2024

A unified structure-preserving parametric finite element method for anisotropic surface diffusion

We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions $(d=2)$ and surfaces in three dimensions $(d=3)$ with an arbitrary anisotropic surface energy density $γ(\boldsymbol{n})$, where $\boldsymbol{n}\in \mathbb{S}^{d-1}$ represents the outward unit vector. By introducing a novel unified surface energy matrix $\boldsymbol{G}_k(\boldsymbol{n})$ depending on $γ(\boldsymbol{n})$, the Cahn--Hoffman $\boldsymbolξ$-vector and a stabilizing function $k(\boldsymbol{n}):\ \mathbb{S}^{d-1}\to {\mathbb R}$, we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators including the surface gradient operator, the surface divergence operator and the surface Laplace--Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on $γ(\boldsymbol{n})$, we propose a new framework via {\sl local energy estimate} for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density $γ(\boldsymbol{n})$ arising from different applications.

math.NA↗

arXiv2024

A robust hybridizable discontinuous Galerkin scheme with harmonic averaging technique for steady state of real-world semiconductor devices

Solving real-world nonlinear semiconductor device problems modeled by the drift-diffusion equations coupled with the Poisson equation (also known as the Poisson-Nernst-Planck equations) necessitates an accurate and efficient numerical scheme which can avoid non-physical oscillations even for problems with extremely sharp doping profiles. In this paper, we propose a flexible and high-order hybridizable discontinuous Galerkin (HDG) scheme with harmonic averaging (HA) technique to tackle these challenges. The proposed HDG-HA scheme combines the robustness of finite volume Scharfetter-Gummel (FVSG) method with the high-order accuracy and $hp$-flexibility offered by the locally conservative HDG scheme. The coupled Poisson equation and two drift-diffusion equations are simultaneously solved by the Newton method. Indicators based on the gradient of net doping $N$ and solution variables are proposed to switch between cells with HA technique and high-order conventional HDG cells, utilizing the flexibility of HDG scheme. Numerical results suggest that the proposed scheme does not exhibit oscillations or convergence issues, even when applied to heavily doped and sharp PN-junctions. Devices with circular junctions and realistic doping profiles are simulated in two dimensions, qualifying this scheme for practical simulation of real-world semiconductor devices.

math.NA↗

arXiv2024

Convergence rates in the nonrelativistic limit of the cubic Klein-Gordon equation

In this paper, we study the nonrelativistic limit of the cubic nonlinear Klein-Gordon equation in $\mathbb{R}^{3}$ with a small parameter $0<\varepsilon \ll 1$, which is inversely proportional to the speed of light. We show that the cubic nonlinear Klein-Gordon equation converges to the cubic nonlinear Schrödinger equation with a convergence rate of order $\varepsilon^{2}$. In particular, for the defocusing case and smooth initial data, we prove error estimates of the form $(1+t)\varepsilon^{2}$ at time $t$ which is valid up to long time of order $\varepsilon^{-1}$; while for nonsmooth initial data, we prove error estimates of the form $(1+t)\varepsilon$ at time $t$ which is valid up to long time of order $\varepsilon^{-\frac{1}{2}}$. These specific forms of error estimates coincide with the numerical results obtained in \cite{BZ19,SZ20}.

math.AP↗

arXiv2024

An explicit and symmetric exponential wave integrator for the nonlinear Schrödinger equation with low regularity potential and nonlinearity

We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schrödinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form $ |ψ|^{2σ}ψ$ with $ ψ$ being the wave function and $ σ> 0 $ being the exponent of the nonlinearity. The sEWI is explicit and stable under a time step size restriction independent of the mesh size. We rigorously establish error estimates of the sEWI under various regularity assumptions on potential and nonlinearity. For ``good" potential and nonlinearity ($H^2$-potential and $σ\geq 1$), we establish an optimal second-order error bound in the $L^2$-norm. For low regularity potential and nonlinearity ($L^\infty$-potential and $σ> 0$), we obtain a first-order $L^2$-norm error bound accompanied with a uniform $H^2$-norm bound of the numerical solution. Moreover, adopting a new technique of \textit{regularity compensation oscillation} (RCO) to analyze error cancellation, for some non-resonant time steps, the optimal second-order $L^2$-norm error bound is proved under a weaker assumption on the nonlinearity: $σ\geq 1/2$. For all the cases, we also present corresponding fractional order error bounds in the $H^1$-norm, which is the natural norm in terms of energy. Extensive numerical results are reported to confirm our error estimates and to demonstrate the superiority of the sEWI, including much weaker regularity requirements on potential and nonlinearity, and excellent long-time behavior with near-conservation of mass and energy.

math.NA↗

arXiv2024

An energy-stable parametric finite element method for the planar Willmore flow

We propose an energy-stable parametric finite element method (PFEM) for the planar Willmore flow and establish its unconditional energy stability of the full discretization scheme. The key lies in the introduction of two novel geometric identities to describe the planar Willmore flow: the first one involves the coupling of the outward unit normal vector $\boldsymbol{n}$ and the normal velocity $V$, and the second one concerns the time derivative of the mean curvature $κ$. Based on them, we derive a set of new geometric partial differential equations for the planar Willmore flow, leading to our new fully-discretized and unconditionally energy-stable PFEM. Our stability analysis is also based on the two new geometric identities. Extensive numerical experiments are provided to illustrate its efficiency and validate its unconditional energy stability.

math.NA↗