Related papers: Solving time-dependent PDEs with the ultraspherica…
In this work, we develop a localized numerical scheme with low regularity requirements for solving time-fractional integro-differential equations. First, a fully discrete numerical scheme is constructed. Specifically, for temporal…
In this paper we are concerned with Trefftz discretizations of the time-dependent linear wave equation in anisotropic media in arbitrary space dimensional domains $\Omega \subset \mathbb{R}^d~ (d\in \mathbb{N})$. We propose two variants of…
A class of abstract nonlinear time-periodic evolution problems is considered which arise in electrical engineering and other scientific disciplines. An efficient solver is proposed for the systems arising after discretization in time based…
This paper presents a novel approach for numerical solution of a class of fourth order time fractional partial differential equations (PDE's). The finite difference formulation has been used for temporal discretization, whereas, the space…
Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient…
It is found that the wave functions of the Gross-Pitaevskii equation (GPE) often vary significantly in different spatial regions, with some components exhibiting sharp variations while others remain smooth. Solving the GPE on a single mesh,…
We show that a pseudospectral representation of the wavefunction using multiple spatial domains of variable size yields a highly accurate, yet efficient method to solve the time-dependent Schr\"odinger equation. The overall spatial domain…
We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr\"odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for…
This work continues a line of works on developing partially explicit methods for multiscale problems. In our previous works, we have considered linear multiscale problems, where the spatial heterogeneities are at subgrid level and are not…
We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time. Our scheme combines asymptotic techniques (which are inexpensive but can have insufficient accuracy) with parallel-in-time methods (which,…
A high-order convergent numerical method for solving linear and non-linear parabolic PDEs is presented. The time-stepping is done via an explicit, singly diagonally implicit Runge-Kutta (ESDIRK) method of order 4 or 5, and for the implicit…
The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method,to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The…
This paper presents a novel approach to rigorously solving initial value problems for semilinear parabolic partial differential equations (PDEs) using fully spectral Fourier-Chebyshev expansions. By reformulating the PDE as a system of…
We present a numerical spectral method to solve systems of differential equations on an infinite interval $y\in (-\infty, \infty)$ in presence of linear differential operators of the form $Q(y) \left(\partial/\partial_y\right)^b$ (where…
A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential…
In the task of predicting spatio-temporal fields in environmental science using statistical methods, introducing statistical models inspired by the physics of the underlying phenomena that are numerically efficient is of growing interest.…
This paper is concerned with developing and analyzing two novel implicit temporal discretization methods for the stochastic semilinear wave equations with multiplicative noise. The proposed methods are natural extensions of well-known…
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains…
Spectral methods provide highly accurate numerical solutions for partial differential equations, exhibiting exponential convergence with the number of spectral nodes. Traditionally, in addressing time-dependent nonlinear problems, attention…
We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled forward--backward SDEs, which provides an efficient probabilistic representation of this type of equation.…