arXiv subjects

Yongyong Cai

Publications and source records attributed to Yongyong Cai.

At least 19 recordsLinked to original sources

arXiv2026

Unconditional Optimal Error Estimates and Energy Stability for a Linearly Implicit Mass-Lumped Projection Finite Element Method for the Harmonic Map Flow

We propose and analyze a linearly implicit mass-lumped finite element method for the heat flow of harmonic maps into the unit sphere. The method consists of a linear predictor followed by a nodal projection and therefore preserves the unit-length constraint exactly at all finite element nodes. The predictor is derived from a cross-product reformulation of the equation and is shown to be equivalent to a mass-lumped discretization of the original formulation with a correction term enforcing nodal orthogonality, as well as to a tangent plane scheme. A key ingredient is the consistent use of the discrete inner product in both the mass and stiffness terms. This yields a nodal orthogonality relation implying that the auxiliary solution lies on or outside the unit sphere at every node. Consequently, the projection is well defined and the projected error satisfies a contraction property in the discrete $L^2$-norm. On Cartesian rectangular and cuboidal tensor-product meshes, the nodal projection is also nonexpansive in a discrete Dirichlet energy, which gives an unconditional discrete energy dissipation law. For sufficiently smooth solutions, we prove optimal error estimates without any coupling condition between the time step and the mesh size: the method converges with order $O(Δt+h^2)$ in $\ell^\infty(0,T;L^2)$ and order $O(Δt+h)$ in $\ell^2(0,T;H^1)$. The proof combines the projected-error contraction, quadrature consistency estimates, edge-based cancellation identities, and a bootstrap argument for controlling nonlinear terms. Numerical experiments confirm the predicted convergence rates and the discrete energy decay.

math.NA↗

arXiv2026

Tensor gradient flow for rod-like liquid crystals from molecular model with closure approximation by quasi-entropy

In tensor dynamics for liquid crystals derived from molecular models, a common problem is closure approximation. For rod-like molecules, the Bingham closure has proved to outperform other methods because it inherits the gradient flow structure of the molecular model, but is difficult to achieve efficient computations maintaining the gradient flow structure. We propose a closure approximation by the quasi-entropy that has been successfully applied to the free energy, based on which we construct the tensor gradient flow. The quasi-entropy closure has the same symmetry properties as the Bingham closure. The resulting tensor gradient flow is able to constrain the eigenvalues of the tensor within the physical range, guaranteeing the positive definiteness of the dissipation operator given by the higher-order tensors. The quasi-entropy closure is easy to implement since it can be reduced to minimizing an elementary function of three variables. As a result, we construct a numerical scheme preserving the eigenvalue constraints and energy dissipation, with the closure approximation decoupled from solving the scheme. Numerical simulations are carried out for the interface between the isotropic and the uniaxial nematic phase, as well as the defect evolutions, where the higher-order tensors indeed make a difference.

cond-mat.soft↗

arXiv2025

Mathematical and numerical studies on ground states of trapped unitary Fermi gases

We mathematically and numerically study the ground states of unitary Fermi gases. Starting from the three-dimensional nonlinear Schrödinger equation that contains a quantum pressure term and an angular momentum rotation term, we first nondimensionalize the equation and then obtain its one-dimensional and two-dimensional counterparts in some limit regimes of the external potentials. Existence and uniqueness of the ground states of the unitary Fermi gases are studied with/without the angular momentum rotation term. We present a regularized normalized gradient flow method to compute the ground states of trapped unitary Fermi gases. Our numerical results show that the quantum pressure term has a significant effect on the ground state properties. Specifically, with the presence of the quantum pressure term, the vortex lattices are very different from those obtained in conventional Bose-Einstein condensation.

math-ph↗

arXiv2025

Efficient and stable diffusion generated methods for ground state computation in Bose--Einstein condensates

This paper investigates numerical methods for approximating the ground state of Bose--Einstein condensates (BECs) by introducing two relaxed formulations of the Gross--Pitaevskii energy functional. These formulations achieve first- and second-order accuracy with respect to the relaxation parameter $ τ$, and are shown to converge to the original energy functional as $ τ\to 0 $. A key feature of the relaxed functionals is their concavity, which ensures that local minima lie on the boundary of the concave hull. This property prevents energy increases during constraint normalization and enables the development of energy-dissipative algorithms. Numerical methods based on sequential linear programming are proposed, accompanied by rigorous analysis of their stability with respect to the relaxed energy. To enhance computational efficiency, an adaptive strategy is introduced, dynamically refining solutions obtained with larger relaxation parameters to achieve higher accuracy with smaller ones. Numerical experiments demonstrate the stability, convergence, and energy dissipation of the proposed methods, while showcasing the adaptive strategy's effectiveness in improving computational performance.

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

Error estimates of a regularized finite difference method for the Logarithmic Schrödinger equation with Dirac delta potential

In this paper, we introduce a conservative Crank-Nicolson-type finite difference schemes for the regularized logarithmic Schrödinger equation (RLSE) with Dirac delta potential in 1D. The regularized logarithmic Schrödinger equation with a small regularized parameter $0<\eps \ll 1$ is adopted to approximate the logarithmic Schrödinger equation (LSE) with linear convergence rate $O(\eps)$. The numerical method can be used to avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in LSE. Then, by using domain-decomposition technique, we can transform the original problem into an interface problem. Different treatments on the interface conditions lead to different discrete schemes and it turns out that a simple discrete approximation of the Dirac potential coincides with one of the conservative finite difference schemes. The optimal $H^1$ error estimates and the conservative properties of the finite difference schemes are investigated. The Crank-Nicolson finite difference methods enjoy the second-order convergence rate in time and space. Numerical examples are provided to support our analysis and show the accuracy and efficiency of the numerical method.

math.NA↗

arXiv2024

Positivity preserving and mass conservative projection method for the Poisson-Nernst-Planck equation

We propose and analyze a novel approach to construct structure preserving approximations for the Poisson-Nernst-Planck equations, focusing on the positivity preserving and mass conservation properties. The strategy consists of a standard time marching step with a projection (or correction) step to satisfy the desired physical constraints (positivity and mass conservation). Based on the $L^2$ projection, we construct a second order Crank-Nicolson type finite difference scheme, which is linear (exclude the very efficient $L^2$ projection part), positivity preserving and mass conserving. Rigorous error estimates in $L^2$ norm are established, which are both second order accurate in space and time. The other choice of projection, e.g. $H^1$ projection, is discussed. Numerical examples are presented to verify the theoretical results and demonstrate the efficiency of the proposed method.

math.NA↗

arXiv2023

Revealing Excited States of Rotational Bose-Einstein Condensates

Rotational Bose-Einstein condensates can exhibit quantized vortices as topological excitations. In this study, the ground and excited states of the rotational Bose-Einstein condensates are systematically studied by calculating the stationary points of the Gross-Pitaevskii energy functional. Various excited states and their connections at different rotational frequencies are revealed in solution landscapes constructed with the constrained high-index saddle dynamics method. Four excitation mechanisms are identified: vortex addition, rearrangement, merging, and splitting. We demonstrate changes in the ground state with increasing rotational frequencies and decipher the evolution of the stability of ground states.

cond-mat.quant-gas↗

arXiv2022

Stabilized exponential time differencing schemes for the convective Allen-Cahn equation

The convective Allen-Cahn equation has been widely used to simulate multi-phase flows in many phase-field models. As a generalized form of the classic Allen-Cahn equation, the convective Allen-Cahn equation still preserves the maximum bound principle (MBP) in the sense that the time-dependent solution of the equation with appropriate initial and boundary conditions preserves for all time a uniform pointwise bound in absolute value. In this paper, we develop efficient first- and second-order exponential time differencing (ETD) schemes combined with the linear stabilizing technique to preserve the MBP unconditionally in the discrete setting. The space discretization is done using the upwind difference scheme for the convective term and the central difference scheme for the diffusion term, and both the mobility and nonlinear terms are approximated through the linear convex interpolation. The unconditional preservation of the MBP of the proposed schemes is proven, and their convergence analysis is presented. Various numerical experiments in two and three dimensions are also carried out to verify the theoretical results.

math.NA↗

arXiv2022

Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schrödinger equation

We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schrödinger equation with small potential and the nonlinear Schrödinger equation (NLSE) with weak nonlinearity. For the Schrödinger equation with small potential characterized by a dimensionless parameter $\varepsilon \in (0, 1]$ representing the amplitude of the potential, we employ the unitary flow property of the (second-order) time-splitting Fourier pseudospectral (TSFP) method in $L^2$-norm to prove a uniform error bound at $C(T)(h^m +τ^2)$ up to the long time $T_\varepsilon= T/\varepsilon$ for any $T>0$ and uniformly for $0<\varepsilon\le1$, while $h$ is the mesh size, $τ$ is the time step, $m \ge 2$ depends on the regularity of the exact solution, and $C(T) =C_0+C_1T$ grows at most linearly with respect to $T$ with $C_0$ and $C_1$ two positive constants independent of $T$, $\varepsilon$, $h$ and $τ$. Then by introducing a new technique of {\sl regularity compensation oscillation} (RCO) in which the high frequency modes are controlled by regularity and the low frequency modes are analyzed by phase cancellation and energy method, an improved uniform error bound at $O(h^{m-1} + \varepsilon τ^2)$ is established in $H^1$-norm for the long-time dynamics up to the time at $O(1/\varepsilon)$ of the Schrödinger equation with $O(\varepsilon)$-potential with $m \geq 3$, which is uniformly for $\varepsilon\in(0,1]$. Moreover, the RCO technique is extended to prove an improved uniform error bound at $O(h^{m-1} + \varepsilon^2τ^2)$ in $H^1$-norm for the long-time dynamics up to the time at $O(1/\varepsilon^2)$ of the cubic NLSE with $O(\varepsilon^2)$-nonlinearity strength, uniformly for $\varepsilon \in (0, 1]$. Extensions to the first-order and fourth-order time-splitting methods are discussed.

math.NA↗

arXiv2022

Improved uniform error bounds on time-splitting methods for the long-time dynamics of the weakly nonlinear Dirac equation

Improved uniform error bounds on time-splitting methods are rigorously proven for the long-time dynamics of the weakly nonlinear Dirac equation (NLDE), where the nonlinearity strength is characterized by a dimensionless parameter $\varepsilon \in (0, 1]$ . We adopt a second order Strang splitting method to discretize the NLDE in time and combine the Fourier pseudospectral method in space for the full-discretization. By employing the {\sl regularity compensation oscillation} (RCO) technique where the high frequency modes are controlled by the regularity of the exact solution and the low frequency modes are analyzed by phase cancellation and energy method, we establish improved uniform error bounds at $O(\varepsilon^2τ^2)$ and $O(h^{m-1}+ \varepsilon^2τ^2)$ for the second-order Strang splitting semi-discretizaion and full-discretization up to the long-time $T_{\varepsilon} = T/\varepsilon^2$ with $T>0$ fixed, respectively. Furthermore, the numerical scheme and error estimates are extended to an oscillatory NLDE which propagates waves with $O(\varepsilon^2)$ wavelength in time. Finally, numerical examples verifying our analytical results are given.

math.NA↗

arXiv2021

Convergence Analysis of A Second-order Accurate, Linear Numerical Scheme for The Landau-Lifshitz Equation with Large Damping Parameters

A second order accurate, linear numerical method is analyzed for the Landau-Lifshitz equation with large damping parameters. This equation describes the dynamics of magnetization, with a non-convexity constraint of unit length of the magnetization. The numerical method is based on the second-order backward differentiation formula in time, combined with an implicit treatment of the linear diffusion term and explicit extrapolation for the nonlinear terms. Afterward, a projection step is applied to normalize the numerical solution at a point-wise level. This numerical scheme has shown extensive advantages in the practical computations for the physical model with large damping parameters, which comes from the fact that only a linear system with constant coefficients (independent of both time and the updated magnetization) needs to be solved at each time step, and has greatly improved the numerical efficiency. Meanwhile, a theoretical analysis for this linear numerical scheme has not been available. In this paper, we provide a rigorous error estimate of the numerical scheme, in the discrete $\ell^{\infty}(0,T; \ell^2) \cap \ell^2(0,T; H_h^1)$ norm, under suitable regularity assumptions and reasonable ratio between the time step-size and the spatial mesh-size. In particular, the projection operation is nonlinear, and a stability estimate for the projection step turns out to be highly challenging. Such a stability estimate is derived in details, which will play an essential role in the convergence analysis for the numerical scheme, if the damping parameter is greater than 3.

math.NA↗

arXiv2021

Improved uniform error bounds on time-splitting methods for long-time dynamics of the nonlinear Klein--Gordon equation with weak nonlinearity

We establish improved uniform error bounds on time-splitting methods for the long-time dynamics of the nonlinear Klein--Gordon equation (NKGE) with weak cubic nonlinearity, whose strength is characterized by $\varepsilon^2$ with $0 < \varepsilon \leq 1$ a dimensionless parameter. Actually, when $0 < \varepsilon \ll 1$, the NKGE with $O(\varepsilon^2)$ nonlinearity and $O(1)$ initial data is equivalent to that with $O(1)$ nonlinearity and small initial data of which the amplitude is at $O(\varepsilon)$. We begin with a semi-discretization of the NKGE by the second-order time-splitting method, and followed by a full-discretization via the Fourier spectral method in space. Employing the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the exact solution and analyzes the low frequency modes by phase cancellation and energy method, we carry out the improved uniform error bounds at $O(\varepsilon^2τ^2)$ and $O(h^m+\varepsilon^2τ^2)$ for the second-order semi-discretization and full-discretization up to the long time $T_\varepsilon = T/\varepsilon^2$ with $T$ fixed, respectively. Extensions to higher order time-splitting methods and the case of an oscillatory complex NKGE are also discussed. Finally, numerical results are provided to confirm the improved error bounds and to demonstrate that they are sharp.

math.NA↗

arXiv2021

A second-order numerical method for Landau-Lifshitz-Gilbert equation with large damping parameters

A second order accurate numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method are associated with the following features: (1) It only solves linear systems of equations with constant coefficients where fast solvers are available, so that the numerical efficiency has been greatly improved, in comparison with the existing Gauss-Seidel project method. (2) The second-order accuracy in time is achieved, and it is unconditionally stable for large damping parameters. Moreover, both the second-order accuracy and the great efficiency improvement will be verified by several numerical examples in the 1D and 3D simulations. In the presence of large damping parameters, it is observed that this method is unconditionally stable and finds physically reasonable structures while many existing methods have failed. For the domain wall dynamics, the linear dependence of wall velocity with respect to the damping parameter and the external magnetic field will be obtained through the reported simulations.

physics.comp-ph↗

arXiv2021

Uniform error bounds of time-splitting methods for the nonlinear Dirac equation in the nonrelativistic regime without magnetic potential

Super-resolution of the Lie-Trotter splitting ($S_1$) and Strang splitting ($S_2$) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic regime with a small parameter $0<\varepsilon\leq 1$ inversely proportional to the speed of light. In this regime, the solution highly oscillates in time with wavelength at $O(\varepsilon^2)$. The splitting methods surprisingly show super-resolution, i.e. the methods can capture the solution accurately even if the time step size $τ$ is much larger than the sampled wavelength at $O(\varepsilon^2)$. Similar to the linear case, $S_1$ and $S_2$ both exhibit $1/2$ order convergence uniformly with respect to $\varepsilon$. Moreover, if $τ$ is non-resonant, i.e. $τ$ is away from certain region determined by $\varepsilon$, $S_1$ would yield an improved uniform first order $O(τ)$ error bound, while $S_2$ would give improved uniform $3/2$ order convergence. Numerical results are reported to confirm these rigorous results. Furthermore, we note that super-resolution is still valid for higher order splitting methods.

math.NA↗

arXiv2020

Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime

We establish error bounds of the Lie-Trotter splitting ($S_1$) and Strang splitting ($S_2$) for the Dirac equation in the nonrelativistic limit regime in the absence of external magnetic potentials, with a small parameter $0<\varepsilon\leq 1$ inversely proportional to the speed of light. In this limit regime, the solution propagates waves with $O(\varepsilon^2)$ wavelength in time. Surprisingly, we find out that the splitting methods exhibit super-resolution, in the sense of breaking the resolution constraint under the Shannon's sampling theorem, i.e. the methods can capture the solutions accurately even if the time step size $τ$ is much larger than the sampled wavelength at $O(\varepsilon^2)$. $S_1$ shows $1/2$ order convergence uniformly with respect to $\varepsilon$, by establishing that there are two independent error bounds $τ+ \varepsilon$ and $τ+ τ/\varepsilon$. Moreover, if $τ$ is non-resonant, i.e. $τ$ is away from certain region determined by $\varepsilon$, $S_1$ would yield an improved uniform first order $O(τ)$ error bound. In addition, we show $S_2$ is uniformly convergent with 1/2 order rate for general time step size $τ$ and uniformly convergent with $3/2$ order rate for non-resonant time step size. Finally, numerical examples are reported to validate our findings.

math.NA↗

arXiv2019

Ground states and their characterization of spin-$F$ Bose-Einstein condensates

The computation of the ground states of spin-$F$ Bose-Einstein condensates (BECs) can be formulated as an energy minimization problem with two quadratic constraints. We discretize the energy functional and constraints using the Fourier pseudospectral schemes and view the discretized problem as an optimization problem on manifold. Three different types of retractions to the manifold are designed. They enable us to apply various optimization methods on manifold to solve the problem. Specifically, an adaptive regularized Newton method is used together with a cascadic multigrid technique to accelerate the convergence. According to our limited knowledege, our method is the first applicable algorithm for BECs with an arbitrary integer spin, including the complicated spin-3 BECs. Extensive numerical results on ground states of spin-1, spin-2 and spin-3 BECs with diverse interaction and optical lattice potential in one/two/three dimensions are reported to show the efficiency of our method and to demonstrate some interesting physical phenomena.

math.NA↗

arXiv2018

Ground states of Bose-Einstein condensates with higher order interaction

We analyze the ground state of a Bose-Einstein condensate in the presence of higher-order interaction (HOI), modeled by a modified Gross-Pitaevskii equation (MGPE). In fact, due to the appearance of HOI, the ground state structures become very rich and complicated. We establish the existence and non-existence results under different parameter regimes, and obtain their limiting behaviors and/or structures with different combinations of HOI and contact interactions. Both the whole space case and the bounded domain case are considered, where different structures of ground states are identified.

math-ph↗