An Euler-type method for Volterra integro-differential equations

TL;DR

The paper tackles solving Volterra integro-differential equations with diverse kernel structures by introducing an explicit Euler-type single-step scheme augmented with a composite Trapezium Rule to approximate the integral. Richardson extrapolation is employed to dramatically improve accuracy, achieving better than $10^{-12}$ in the numerical examples. For the $n=1$ case, the update is $y_{i+1} = y_i + h f(x_i,y_i) + \frac{h^2}{2}\left(\sum_{j=0}^{i} 2K_j - (K_0+K_i)\right)$, with extensions to higher $n$ and dependent kernels provided. The results demonstrate a simple, fast, and robust method suitable as an accessible alternative to more complex solvers, while noting stability analyses and handling of nonseparable or weakly singular kernels as directions for future work.

Abstract

We describe an algorithm, based on Euler's method, for solving Volterra integro-differential equations. The algorithm approximates the relevant integral by means of the composite Trapezium Rule, using the discrete nodes of the independent variable as the required nodes for the integration variable. We have developed an error control device, using Richardson extrapolation, and we have achieved accuracy better than 1e-12 for all numerical examples considered.