Suppose I have an integro-differential equation of the form
$$y'(x) = f(x,y(x)) + \int_{x_0}^{x} ds\; F(x,s,y(s))$$
With some initial condition $y(x_0) = y_0$. The paper "Note on the Numerical Solution in Integer-Differential Equations" by James Thomas Day presents a numerical integration algorithm that is a simple extension of Euler's method.
In some physical applications, it is useful to have a time-reversal invariant (symplectic) numerical integration algorithm. For example, when numerically solving the orbit of a satellite, the time-reversal invariant leap frog algorithm gives stable, closed orbits while Euler's method does not.
Is there an analog of the leap frog algorithm for integro-differential equations? That is, is there a way to modify the simple algorithm presented by James Thomas Day to make it time-reversal invariant?
Citations:
- James Thomas Day, Note on the Numerical Solution of Integro-Differential Equations, The Computer Journal, Volume 9, Issue 4, February 1967, Pages 394–395, https://doi.org/10.1093/comjnl/9.4.394