Related papers: Symmetric-conjugate splitting methods for linear u…

Nonlinear parabolic equations are frequently encountered in applications and efficient approximating techniques for their solution are of great importance. In order to provide an effective scheme for the temporal approximation of such…

The filtered Lie splitting scheme is an established method for the numerical integration of the periodic nonlinear Schr\"{o}dinger equation at low regularity. Its temporal convergence was recently analyzed in a framework of discrete…

A space-time adaptive method is presented for the reactive Euler equations describing chemically reacting gas flow where a two species model is used for the chemistry. The governing equations are discretized with a finite volume method and…

We develop a unified approach for construction of symplectic forms for 1D integrable equations with the periodic and rapidly decaying initial data. As an example we consider the cubic nonlinear Schr\"{o}dinger equation.

An integrable hierarchies connected with linear stationary Schr\"odinger equation with energy dependent potentials (in general case) are considered. Galilei-like and scaling invariance transformations are constructed. A symmetry method is…

We present new approaches for solving constrained multicomponent nonlinear Schr\"odinger equations in arbitrary dimensions. The idea is to introduce an artificial time and solve an extended damped second order dynamic system whose…

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…

In this paper, we develop a novel class of linear energy-preserving integrating factor methods for the 2D nonlinear Schr\"odinger equation with wave operator (NLSW), combining the scalar auxiliary variable approach and the integrating…

We propose a new primal-dual splitting method for solving composite inclusions involving Lipschitzian, and parallel-sum-type monotone operators. Our approach extends the framework in \cite{Siopt4} to a more general class of monotone…

In this paper, we develop a numerical algorithm for an inverse problem on determining fractional orders of time derivatives simultaneously in a coupled subdiffusion system. Following the theoretical uniqueness, we reformulate the order…

Trigonometric time integrators are introduced as a class of explicit numerical methods for quasilinear wave equations. Second-order convergence for the semi-discretization in time with these integrators is shown for a sufficiently regular…

We show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The argument is based on modular techniques combined with a…

We propose a new approach to the combinatorial interpretations of linearization coefficient problem of orthogonal polynomials. We first establish a difference system and then solve it combinatorially and analytically using the method of…

Most numerical methods for time integration use real time steps. Complex time steps provide an additional degree of freedom, as we can select the magnitude of the step in both the real and imaginary directions. By time stepping along…

Using the transformations from paper I, we show that the Schr\"odinger equations for: (1)systems described by quadratic Hamiltonians, (2) systems with time-varying mass, and (3) time-dependent oscillators, all have isomorphic Lie space-time…

We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since…

We analyse a splitting integrator for the time discretization of the Schr\"odinger equation with nonlocal interaction cubic nonlinearity and white noise dispersion. We prove that this time integrator has order of convergence one in the…

We study the Strang splitting scheme for quasilinear Schr\"odinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the…

We introduce some general tools to design exact splitting methods to compute numerically semigroups generated by inhomogeneous quadratic differential operators. More precisely, we factorize these semigroups as products of semigroups that…

We prove for the first time that, if a linear inverse problem exhibits a group symmetry structure, gradient-based optimizers can be designed to exploit this structure for faster convergence rates. This theoretical finding demonstrates the…