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First-order exponential integrator method

Exponential integrators describe a large class of integrators that have recently seen a lot of development. They date back to at least the 1960s.

In place of (1), we assume the differential equation is either of the form

or it has been locally linearized about a background state to produce a linear term − A y {\displaystyle -Ay} and a nonlinear term N (y) {\displaystyle {\mathcal {N}}(y)}.

Exponential integrators are constructed by multiplying (7) by e A t {\textstyle e^{At}}, and exactly integrating the result overa time interval {\displaystyle }:

This integral equation is exact, but it doesn't define the integral.

The first-order exponential integrator can be realized by holding N (y (t n + τ)) {\displaystyle {\mathcal {N}}(y(t_{n}+\tau))} constant over the full interval: