DeepAI AI Chat
Log In Sign Up

Explicit exponential Runge-Kutta methods for semilinear integro-differential equations

06/12/2022
by   Alexander Ostermann, et al.
0

The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our error bounds for integro-differential equations. The order conditions are further used for constructing numerical methods. The convergence analysis is performed in a Hilbert space setting, where the smoothing effect of the resolvent family is heavily used. For the linear case, we derive the order conditions for general order p and prove convergence of order p, whenever these conditions are satisfied. In the semilinear case, we consider in addition spatial discretization by a spectral Galerkin method, and we require locally Lipschitz continuous nonlinearities. We derive the order conditions for orders one and two, construct methods satisfying these conditions and prove their convergence. Finally, some numerical experiments illustrating our theoretical results are given.

READ FULL TEXT

page 1

page 2

page 3

page 4

10/30/2020

Positivity preserving logarithmic Euler-Maruyama type scheme for stochastic differential equations

In this paper, we propose a class of explicit positivity preserving nume...
02/05/2021

Characterizing Order of Convergence in the Obreshkov Method in Differential-Algebraic Equations

The Obreshkov method is a single-step multi-derivative method used in th...
02/27/2020

Discrete Adjoint Implicit Peer Methods in Optimal Control

It is well known that in the first-discretize-then-optimize approach in ...
11/18/2020

Continuous Galerkin Schemes for Semi-Explicit Differential-Algebraic Equations

This paper studies a new class of integration schemes for the numerical ...
02/25/2022

An ultraweak variational method for parameterized linear differential-algebraic equations

We investigate an ultraweak variational formulation for (parameterized) ...
04/19/2023

Weak Convergence Of Tamed Exponential Integrators for Stochastic Differential Equations

We prove weak convergence of order one for a class of exponential based ...