Related papers: Time integrators for dispersive equations in the l…
Two families of symplectic methods specially designed for second-order time-dependent linear systems are presented. Both are obtained from the Magnus expansion of the corresponding first-order equation, but otherwise they differ in…
This paper presents a solution to an initial value problem for the 1-dimensional wave equation on time scales through the application of a Fourier transform and its inverse via contour integrals. The time scale of the spatial dimension is…
We propose a fast integrator to a class of dynamical systems with several temporal scales. The proposed method is developed as an extension of the variable step size Heterogeneous Multiscale Method (VSHMM), which is a two-scale integrator…
We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity.…
We construct numerical integrators for Hamiltonian problems that may advantageously replace the standard Verlet time-stepper within Hybrid Monte Carlo and related simulations. Past attempts have often aimed at boosting the order of accuracy…
We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation.…
Transport phenomena play a vital role in various fields of science and engineering. In this work, exact solutions are derived for advection equations with integer- and fractional-order time derivatives and a constant time-delay in the…
In this paper, we present a novel spectral renormalization exponential integrator method for solving gradient flow problems. Our method is specifically designed to simultaneously satisfy discrete analogues of the energy dissipation laws and…
In this and subsequent paper arXiv:1011.5185 we develop a recursive approach for calculating the short-time expansion of the propagator for a general quantum system in a time-dependent potential to orders that have not yet been accessible…
We give a way to recover time-dependent singular coefficients of the wave-equation in the one-dimensional case.
Exponential integrators are a well-known class of time integration methods that have been the subject of many studies and developments in the past two decades. Surprisingly, there have been limited efforts to analyze their stability and…
In this paper, we introduce a class of linear time-varying dynamic-algebraic equations(LTVDAE) of tractability index one on arbitrary time scales. We propose a procedure for the decoupling of the considered class LTVDAE. A projector…
We introduce efficient and robust exponential-type integrators for Klein-Gordon equations which resolve the solution in the relativistic regime as well as in the highly-oscillatory non-relativistic regime without any step-size restriction,…
By recursively solving the underlying Schr\" odinger equation, we set up an efficient systematic approach for deriving analytic expressions for discretized effective actions. With this we obtain discrete short-time propagators for both one…
We address the problem of classification of integrable differential-difference equations in 2+1 dimensions with one/two discrete variables. Our approach is based on the method of hydrodynamic reductions and its generalisation to dispersive…
A new method of solution is proposed for solution of the wave equation in one space dimension with continuously-varying coefficients. By considering all paths along which information arrives at a given point, the solution is expressed as an…
Neural operators have shown promise in solving many types of Partial Differential Equations (PDEs). They are significantly faster compared to traditional numerical solvers once they have been trained with a certain amount of observed data.…
The Riemann-Hilbert problem associated with the integrable PDE is used as a nonlinear transformation of the nearly integrable PDE to the spectral space. The temporal evolution of the spectral data is derived with account for arbitrary…
This article is concerned with a new filtered two-step variational integrator for solving the charged-particle dynamics in a mildly non-uniform moderate or strong magnetic field with a dimensionless parameter $\varepsilon$ inversely…
A numerical integrator is presented that computes a symmetric or skew-symmetric low-rank approximation to large symmetric or skew-symmetric time-dependent matrices that are either given explicitly or are the unknown solution to a matrix…