相关论文: A comparison of boundary correction methods for St…
Convergence of the solutions of nonhomogeneous linear singularly perturbed systems to that of the corresponding reduced singular system on the half-line [0, $\infty $) is considered. To include the situation on a neighborhood of initial…
We consider applying the Strang splitting to semilinear parabolic problems. The key ingredients of the Strang splitting are the decomposition of the equation into several parts and the computation of approximate solutions by combining the…
In this paper we consider splitting methods for nonlinear ordinary differential equations in which one of the (partial) flows that results from the splitting procedure can not be computed exactly. Instead, we insert a well-chosen state…
We analyze operator splitting methods applied to scalar equations with a nonlinear advection operator, and a linear (local or nonlocal) diffusion operator or a linear dispersion operator. The advection velocity is determined from the scalar…
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…
We study an approximation method to solve nonlinear multi-term fractional differential equations with initial conditions or boundary conditions. First, we transform the nonlinear multi-term fractional differential equations with initial…
Reaction-Diffusion systems arise in diverse areas of science and engineering. Due to the peculiar characteristics of such equations, analytic solutions are usually not available and numerical methods are the main tools for approximating the…
We assess the applicability and efficiency of time-adaptive high-order splitting methods applied for the numerical solution of (systems of) nonlinear parabolic problems under periodic boundary conditions. We discuss in particular several…
The boundary integral method is extended to derive closed integro-differential equations applicable to computation of the shape and propagation speed of a steadily moving spot and to the analysis of dynamic instabilities in the sharp…
In this paper we propose a Local Orthogonal Decomposition method (LOD) for elliptic partial differential equations with inhomogeneous Dirichlet- and Neumann boundary conditions. For this purpose, we present new boundary correctors which…
Super-resolution of the Lie-Trotter splitting ($S_1$) and Strang splitting ($S_2$) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic regime with a small parameter…
In this work we study operator splitting methods for a certain class of coupled abstract Cauchy problems, where the coupling is such that one of the problems prescribes a "boundary type" extra condition for the other one. The theory of…
We consider an aggregation model with nonlinear diffusion in domains with boundaries and investigate the zero diffusion limit of its solutions. We establish the convergence of weak solutions for fixed times, as well as the convergence of…
We study the long-time dynamics of the nonlinear processes modeled by diffusion-transport partial differential equations in non-divergence form with drifts. The solutions are subject to some inhomogeneous Dirichlet boundary condition.…
Splitting methods constitute a widely used class of numerical integrators for ordinary and partial differential equations, particularly well suited to problems that can be decomposed into simpler subproblems. High-order splitting schemes…
We establish error bounds of the Lie-Trotter splitting ($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…
We apply the cell merging method to a model shallow water problem with a permeable boundary. We use a cut cell approach which is more easily and systematically scalable with different shapes of boundaries. The novel cell merging method…
Nonconvex and structured optimization problems arise in many engineering applications that demand scalable and distributed solution methods. The study of the convergence properties of these methods is in general difficult due to the…
We study reaction-diffusion equations in cylinders with possibly nonlinear diffusion and possibly nonlinear Neumann boundary conditions. We provide a geometric Poincar\'e-type inequality and classification results for stable solutions, and…
This work proposes and analyzes an efficient numerical method for solving the nonlinear Schr\"odinger equation with quasiperiodic potential, where the projection method is applied in space to account for the quasiperiodic structure and the…