arXiv subjects

Liying Sun

Publications and source records attributed to Liying Sun.

At least 19 recordsLinked to original sources

arXiv2026

Strong error analysis of a temporal approximation for stochastic Korteweg-de Vries equation with small additive noise

We study strong temporal approximation of periodic stochastic Korteweg--de Vries equation driven by small additive $Q$-Wiener noise of amplitude $\mathcal O(\varepsilon)$, $0<\varepsilon\ll1$. Strong error analysis for temporal approximations of stochastic KdV is a challenging problem, due to the additional derivative term in the nonlinearity and thanks to the lack of suitable exponential moment bounds for the exact solutions. Exploiting the small-noise regime, we first decompose the solution into a deterministic KdV flow and a stochastic component; then we linearize the obtained stochastic equation and approximate the resulting equation by means of Fourier analytic techniques. Combining the small-noise linearization error, the discretization error of the linearized equation, and the deterministic temporal approximation error, we prove strong convergence rates of order $\mathcal O(\max(\varepsilon^2,\tau,\varepsilon\tau^{1/2}))$ under $H^1$-regularity and $\mathcal O(\max(\varepsilon^2,\tau))$ under $H^2$-regularity, for the obtained approximation of the original stochastic KdV. To the best of our knowledge, these are the first explicit strong convergence rates shown for numerical time approximations of the stochastic KdV.

math.NA↗

arXiv2026

Novel physical property preserved methods for stochastic Schr\"{o}dinger--KdV equation

In this work, we study the stochastic Schr\"odinger--KdV equation driven by additive noise from both analytical and numerical viewpoints. We first establish the evolution laws for the averaged plasmon number, momentum, and energy, together with the conservation of the averaged particle number. Motivated by these intrinsic structures, we develop two temporal discretizations. One is constructed based on the splitting strategy and Crank--Nicolson scheme, and is shown to preserve the discrete evolution laws of the averaged plasmon number and momentum, as well as the discrete conservation law of the averaged particle number. The other is proposed within the constant scalar auxiliary variable framework, in which the nonlinear energy functional is reformulated so that a modified averaged energy law can be preserved at the discrete level. Combining these temporal discretizations with a local discontinuous Galerkin approximation in space yields structure-preserving full discretizations inheriting the corresponding discrete physical laws. Numerical experiments are presented to validate the theoretical results and to demonstrate the accuracy, robustness, and effectiveness of the proposed methods.

math.NA↗

arXiv2026

Exact controllability of the stochastic Maxwell equation: theory and numerical simulation

This article investigates the exact controllability of three-dimensional stochastic Maxwell equations, a coupled system comprising two stochastic partial differential equations. The research establishes the observability inequality for the backward stochastic Maxwell equations using the multiplier method, and subsequently, proves the exact controllability of the forward equations. The control acting on the diffusion term is found to be indispensable, since exact controllability is destroyed when this control is removed; it is further proved that the controllability result obtained in this paper is achieved with a minimal number of controls. Finally, a numerical algorithm combining a central difference for spatial discretization, a midpoint scheme for temporal discretization and Lagrange multiplier method is proposed, yielding numerical results that offer the control value and lead to deeper insights into the underlying theoretical framework.

math.OC↗

arXiv2026

Splitting AVF method for generalized Langevin equations: probability density function and geometric ergodicity

The generalized Langevin equation (GLE) constitutes a fundamental model for describing nonequilibrium dynamics with memory effects. To overcome the numerical challenges arising from superquadratically growing potentials and degenerate noise, we propose and analyze a structure-preserving splitting averaged vector field (AVF) method for a quasi-Markovian GLE. The core advantage of this method lies in its ability to simultaneously preserve the exponential integrability, Malliavin differentiability, and ergodicity of the underlying continuous system. Notably, by leveraging exponential integrability, Malliavin differentiability, and uniform non-degeneracy of the numerical solution, we obtain the existence and smoothness of its probability density function, which converges to that of the exact solution with first-order accuracy. Furthermore, by validating the Lyapunov condition and the minorization condition using a localized technique, we establish the geometric ergodicity of the numerical solution. Finally, numerical experiments are conducted to confirm the theoretical results.

math.NA↗

arXiv2024

Novel semi-explicit symplectic schemes for nonseparable stochastic Hamiltonian systems

In this manuscript, we propose efficient stochastic semi-explicit symplectic schemes tailored for nonseparable stochastic Hamiltonian systems (SHSs). These semi-explicit symplectic schemes are constructed by introducing augmented Hamiltonians and using symmetric projection. In the case of the artificial restraint in augmented Hamiltonians being zero, the proposed schemes also preserve quadratic invariants, making them suitable for developing semi-explicit charge-preserved multi-symplectic schemes for stochastic cubic Schrödinger equations with multiplicative noise. Through numerical experiments that validate theoretical results, we demonstrate that the proposed stochastic semi-explicit symplectic scheme, which features a straightforward Newton iteration solver, outperforms the traditional stochastic midpoint scheme in terms of effectiveness and accuracy.

math.NA↗

arXiv2023

Polynomial weak approximation for stochastic reaction-diffusion equations near the sharp interface limit

We study weak approximation for stochastic reaction-diffusion equations in the sharp-interface regime, where the diffuse interface thickness $\epsilon$ is small and the dependence of numerical constants on $\epsilon^{-1}$ is a central issue. For an additive-noise stochastic Allen--Cahn type equation, direct stability arguments typically produce weak error bounds with constants growing exponentially in $\epsilon^{-1}$. Such estimates do not capture the polynomial stability expected near the sharp interface limit. We prove polynomial-in-$\epsilon^{-1}$ weak error bounds for an accelerated splitting exponential Euler approximation. The proof combines time-uniform moment estimates, regularity estimates for the exact and numerical dynamics, and time-independent derivative estimates for the Kolmogorov semigroup. The key point is that the averaged regularizing effect of the noise, expressed through asymptotic strong Feller or strong Feller estimates, replaces the deterministic spectral estimate for the linearized Allen--Cahn operator. The result gives a weak approximation theory whose constants depend polynomially on $\epsilon^{-1}$ and explicitly on the covariance regularity and non-degeneracy parameters.

math.NA↗

arXiv2023

Quantifying the effect of random dispersion for logarithmic Schrödinger equation

This paper is concerned with the random effect of the noise dispersion for stochastic logarithmic Schrödinger equation emerged from the optical fibre with dispersion management. The well-posedness of the logarithmic Schrödinger equation with white noise dispersion is established via the regularization energy approximation and a spatial scaling property. For the small noise case, the effect of the noise dispersion is quantified by the proven large deviation principle under additional regularity assumptions on the initial datum. As an application, we show that for the regularized model, the exit from a neighborhood of the attractor of deterministic equation occurs on a sufficiently large time scale. Furthermore, the exit time and exit point in the small noise case, as well as the effect of large noise dispersion, is also discussed for the stochastic logarithmic Schrödinger equation.

math.AP↗

arXiv2023

Novel structure-preserving schemes for stochastic Klein--Gordon--Schrödinger equations with additive noise

Stochastic Klein--Gordon--Schrödinger (KGS) equations are important mathematical models and describe the interaction between scalar nucleons and neutral scalar mesons in the stochastic environment. In this paper, we propose novel structure-preserving schemes to numerically solve stochastic KGS equations with additive noise, which preserve averaged charge evolution law, averaged energy evolution law, symplecticity, and multi-symplecticity. By applying central difference, sine pseudo-spectral method, or finite element method in space and modifying finite difference in time, we present some charge and energy preserved fully-discrete scheme for the original system. In addition, combining the symplectic Runge-Kutta method in time and finite difference in space, we propose a class of multi-symplectic discretizations preserving the geometric structure of the stochastic KGS equation. Finally, numerical experiments confirm theoretical findings.

math.NA↗

arXiv2023

Novel structure-preserving schemes for stochastic Klein--Gordon--Schrödinger equations with additive noise

math.NA↗

arXiv2022

Semi-implicit energy-preserving numerical schemes for stochastic wave equation via SAV approach

In this paper, we propose and analyze semi-implicit numerical schemes for the stochastic wave equation (SWE) with general nonlinearity and multiplicative noise. These numerical schemes, called stochastic scalar auxiliary variable (SAV) schemes, are constructed by transforming the considered SWE into a higher dimensional stochastic system with a stochastic SAV. We prove that they can be solved explicitly and preserve the modified energy evolution law and the regularity structure of the original system. These structure-preserving properties are the keys to overcoming the mutual effect of the noise and nonlinearity. By proving new regularity estimates of the introduced SAV, we establish the strong convergence rate of stochastic SAV schemes and the further fully-discrete schemes with the finite element method in spatial direction. To the best of our knowledge, this is the first result on the construction and strong convergence of semi-implicit energy-preserving schemes for nonlinear SWE.

math.NA↗

arXiv2022

Three kinds of novel multi-symplectic methods for stochastic Hamiltonian partial differential equations

Stochastic Hamiltonian partial differential equations, which possess the multi-symplectic conservation law, are an important and fairly large class of systems. The multi-symplectic methods inheriting the geometric features of stochastic Hamiltonian partial differential equations provide numerical approximations with better numerical stability, and are of vital significance for obtaining correct numerical results. In this paper, we propose three novel multi-symplectic methods for stochastic Hamiltonian partial differential equations based on the local radial basis function collocation method, the splitting technique, and the partitioned Runge-Kutta method. Concrete numerical methods are presented for nonlinear stochastic wave equations, stochastic nonlinear Schrödinger equations, stochastic Korteweg-de Vries equations and stochastic Maxwell equations. We take stochastic wave equations as examples to perform numerical experiments, which indicate the validity of the proposed methods.

math.NA↗

arXiv2021

Structure-preserving splitting methods for stochastic logarithmic Schrödinger equation via regularized energy approximation

In this paper, we study two kinds of structure-preserving splitting methods, including the Lie--Trotter type splitting method and the finite difference type method, for the stochasticlogarithmic Schrödinger equation (SlogS equation) via a regularized energy approximation. We first introduce a regularized SlogS equation with a small parameter $0<ε\ll1$ which approximates the SlogS equation and avoids the singularity near zero density. Then we present a priori estimates, the regularized entropy and energy, and the stochastic symplectic structure of the proposed numerical methods. Furthermore, we derive both the strong convergence rates and the convergence rates of the regularized entropy and energy. To the best of our knowledge, this is the first result concerning the construction and analysis of numerical methods for stochastic Schrödinger equations with logarithmic nonlinearities.

math.NA↗

arXiv2021

Energy-preserving fully-discrete schemes for nonlinear stochastic wave equations with multiplicative noise

In this paper, we focus on constructing numerical schemes preserving the averaged energy evolution law for nonlinear stochastic wave equations driven by multiplicative noise. We first apply the compact finite difference method and the interior penalty discontinuous Galerkin finite element method to discretize space variable and present two semi-discrete schemes, respectively. Then we make use of the discrete gradient method and the Padé approximation to propose efficient fully-discrete schemes. These semi-discrete and fully-discrete schemes are proved to preserve the discrete averaged energy evolution law. In particular, we also prove that the proposed fully-discrete schemes exactly inherit the averaged energy evolution law almost surely if the considered model is driven by additive noise. Numerical experiments are given to confirm theoretical findings.

math.NA↗

arXiv2021

Energy-preserving exponential integrable numerical method for stochastic cubic wave equation with additive noise

In this paper, we present an energy-preserving exponentially integrable numerical method for stochastic wave equation with cubic nonlinearity and additive noise. We first apply the spectral Galerkin method to discretizing the original equation and show that this spatial discretization possesses an energy evolution law and certain exponential integrability property. Then the exponential integrability property of the exact solution is deduced by proving the strong convergence of the semi-discretization. To propose a full discrete numerical method which could inherit both the energy evolution law and the exponential integrability, we use the splitting technique and averaged vector field method in the temporal direction. Combining these structure-preserving properties with regularity estimates of the exact and the numerical solutions, we obtain the strong convergence rate of the numerical method. Numerical experiments coincide with these theoretical results.

math.NA↗

arXiv2021

Stochastic logarithmic Schrodinger equations: energy regularized approach

In this paper, we prove the global existence and uniqueness of the solution of the stochastic logarithmic Schrödinger (SlogS) equation driven by additive noise or multiplicative noise. The key ingredient lies on the regularized stochastic logarithmic Schrödinger (RSlogS) equation with regularized energy and the strong convergence analysis of the solutions of (RSlogS) equations. In addition, temporal Hölder regularity estimates and uniform estimates in energy space $\mathbb H^1(\mathcal O)$ and weighted Sobolev space $L^2_α(\mathcal O)$ of the solutions for both SlogS equation and RSlogS equation are also obtained.

math.PR↗

arXiv2020

Numerically asymptotical preservation of the large deviations principles for invariant measures of Langevin equations

In this paper, we focus on two kinds of large deviations principles (LDPs) of the invariant measures of Langevin equations and their numerical methods, as the noise intensity $ε\to 0$ and the dissipation intensity $ν\to\infty$ respectively. First, by proving the weak LDP and the exponential tightness, we conclude that the invariant measure $\{μ_{ν,ε}\}$ of the exact solution satisfies the LDPs as $ε\to0$ and $ν\to\infty$ respectively. Then, we study whether there exist numerical methods asymptotically preserving these two LDPs of $\{μ_{ν,ε}\}$ in the sense that the rate functions of invariant measures of numerical methods converge pointwise to the rate function of $\{μ_{ν,ε}\}$ as the step-size tends to zero. The answer is positive for the linear Langevin equation. For the small noise case, we show that a large class of numerical methods can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ε>0}$ as $ε\to0$. For the strong dissipation case, we study the stochastic $θ$-method ($θ\in[1/2,1]$) and show that only the midpoint scheme ($θ=1/2$) can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ν>0}$ as $ν\to\infty$. These results indicate that in the linear case, the LDP as $ε\to0$ and the LDP as $ν\to\infty$ for the invariant measures of numerical methods have intrinsic differences: the common numerical methods can asymptotically preserve the LDP of $\{μ_{ν,ε}\}_{ε>0}$ as $ε\to0$ while the asymptotical preservation of numerical methods for the LDP of $\{μ_{ν,ε}\}_{ν>0}$ as $ν\to\infty$ depends on the choice of numerical methods. To the best of our knowledge, this is the first result of investigating the relationship between the LDPs of invariant measures of stochastic differential equations and those of their numerical methods.

math.NA↗

arXiv2020

Numerical analysis of a full discretization for stochastic Cahn--Hilliard equation driven by additive noise

In this article, we consider the stochastic Cahn--Hilliard equation driven by space-time white noise. We discretize this equation by using a spatial spectral Galerkin method and a temporal accelerated implicit Euler method. The optimal regularity properties and uniform moment bounds of the exact and numerical solutions are shown. Then we prove that the proposed numerical method is strongly convergent with the sharp convergence rate in a negative Sobolev space. By using an interpolation approach, we deduce the spatial optimal convergence rate and the temporal super-convergence rate of the proposed numerical method in strong convergence sense. To the best of our knowledge, this is the first result on the strong convergence rates of numerical methods for the stochastic Cahn--Hilliard equation driven by space-time white noise. This interpolation approach is also applied to the general noise and high dimension cases, and strong convergence rate results of the proposed scheme are given.

math.NA↗

arXiv2020

Structure-preserving numerical methods for stochastic Poisson systems

We propose a class of numerical integration methods for stochastic Poisson systems (SPSs) of arbitrary dimensions. Based on the Darboux-Lie theorem, we transform the SPSs to their canonical form, the generalized stochastic Hamiltonian systems (SHSs), via canonical coordinate transformations found by solving certain PDEs defined by the Poisson brackets of the SPSs. An a-generating function approach with α\in [0,1] is then used to create symplectic discretizations of the SHSs, which are then transformed back by the inverse coordinate transformation to numerical integrators for the SPSs. These integrators are proved to preserve both the Poisson structure and the Casimir functions of the SPSs. Applications to a three-dimensional stochastic rigid body system and a three-dimensional stochastic Lotka-Volterra system show efficiency of the proposed methods.

math.NA↗