Exponential Euler and the two-stage ETD2 step maps

Let $d\ge 1$, $A\in M_d(\mathbb{C})$, $g:[0,T]\times U\to {\mathbb{C}}^d$, and $t\ge 0$, $h>0$ with $t+h\le T$. The inputs $y$ and all displayed stages must lie in $U$. The functions ${\phi }_k(hA)$ are the finite-matrix series of The phi-function family by everywhere-convergent series, whose convergence is supplied by The scalar and matrix phi series converge absolutely.

The exponential Euler map is ${\mathrm{\Psi }}_h^E(t,y):=e^{hA}y+h{\phi }_1(hA)g(t,y)\mathrm{.}$

For the two-stage exponential time-differencing Runge–Kutta map ETD2, first form the exponential Euler predictor $a_h(t,y):=e^{hA}y+h{\phi }_1(hA)g(t,y),$ and then set ${\mathrm{\Psi }}_h^2(t,y):=e^{hA}y+h({\phi }_1(hA)-{\phi }_2(hA))g(t,y)+h{\phi }_2(hA)g(t+h,a_h(t,y))\mathrm{.}$ These are exact mathematical step maps; an implementation must separately account for every approximate matrix-function action and arithmetic error.