Related papers: Error estimates of a regularized finite difference…
We propose a locally one dimensional (LOD) finite difference method for multidimensional Riesz fractional diffusion equation with variable coefficients on a finite domain. The numerical method is second-order convergent in both space and…
We construct in this paper global (for $t \geq 0$) and bounded solutions $u(t)$ for the nonlinear Schr\"odinger equation \[i \partial_t u + \Delta u + |u|^{p-1} u = 0, \quad t \in \mathbb{R}, x \in \mathbb{R}^d\] in mass sub-critical cases…
Signal propagation in an optical fiber can be described by the nonlinear Schr\"odinger equation (NLSE). The NLSE has no known closed-form solution, mostly due to the interaction of dispersion and nonlinearities. In this paper, we present a…
We derive optimal order a posteriori error estimates for fully discrete approximations of linear Schr\"odinger-type equations, in the $L^\infty(L^2)-$norm. For the discretization in time we use the Crank-Nicolson method, while for the space…
We address the problem of Schr\"odinger potential estimation, which plays a crucial role in modern generative modelling approaches based on Schr\"odinger bridges and stochastic optimal control for SDEs. Given a simple prior diffusion…
We present several numerical methods and establish their error estimates for the discretization of the nonlinear Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter $0<\varepsilon\ll 1$ which is…
We consider numerical instability that can be observed in simulations of localized solutions of the generalized nonlinear Schr\"odinger equation (NLS) by a split-step method where the linear part of the evolution is solved by a…
This work proposes and analyzes a fully discrete numerical scheme for solving the Landau-Lifshitz-Gilbert (LLG) equation, which achieves fourth-order spatial accuracy and third-order temporal accuracy.Spatially, fourth-order accuracy is…
In this paper, we consider the following logarithmic Schr\"odinger equation \[ -\Delta u + V(x)u = u \log u^{2},\quad x\in\mathbb{R}^{N}. \] Assuming that \(V\in C(\mathbb{R}^{N},\mathbb R)\), \(V\) is bounded away from zero, and…
We propose and analyze a novel symmetric Gautschi-type exponential wave integrator (sEWI) for the nonlinear Schr\"odinger equation (NLSE) with low regularity potential and typical power-type nonlinearity of the form $ |\psi|^{2\sigma}\psi $…
We discuss the finite-time collapse, also referred as blow-up, of the solutions of a discrete nonlinear Schr\"{o}dinger (DNLS) equation incorporating linear and nonlinear gain and loss. This DNLS system appears in many inherently discrete…
The numerical approximation of low-regularity solutions to the nonlinear Schr\"odinger equation is notoriously difficult and even more so if structure-preserving schemes are sought. Recent works have been successful in establishing…
A fully discrete and fully explicit low-regularity integrator is constructed for the one-dimensional periodic cubic nonlinear Schr\"odinger equation. The method can be implemented by using fast Fourier transform with $O(N\ln N)$ operations…
This paper is devoted to order-one explicit approximations of random periodic solutions to multiplicative noise driven stochastic differential equations (SDEs) with non-globally Lipschitz coefficients. The existence of the random periodic…
Numerical homogenization methods aim at providing appropriate coarse-scale approximations of solutions to (elliptic) partial differential equations that involve highly oscillatory coefficients. The localized orthogonal decomposition (LOD)…
The simplified lattice Boltzmann method (SLBM) is a recent development in the lattice Boltzmann method (LBM) community, addressing the intrinsic limitations of the traditional LBM by directly evolving macroscopic quantities and maintaining…
We study the stability of an inverse problem for the fractional conductivity equation on bounded smooth domains. We obtain a logarithmic stability estimate for the inverse problem under suitable a priori bounds on the globally defined…
In this paper, we consider an initial-boundary value problem for the following mixed pseudo-parabolic $p(.)$-Laplacian type equation with logarithmic nonlinearity: $$ u_t-\Delta u_t-\mbox{div}\left(\left\vert \nabla…
One-class classification (OCC) aims to train a classifier only with the target class data and attracts great attention for its strong applicability in real-world application. Despite a lot of advances have been made in OCC, it still lacks…
Some prominent discretisation methods such as finite elements provide a way to approximate a function of $d$ variables from $n$ values it takes on the nodes $x_i$ of the corresponding mesh. The accuracy is $n^{-s_a/d}$ in $L^2$-norm, where…