Future work

• precision on parameter.
• exponential runge-kutta method

The "classical" Runge-Kutta method (order 4) adapted to the exponential (not yet implemented)

Notations :

• We denote by $G$ the function which transforms $\hat{U}$ to $\hat{f}$, so $G(\hat{U}) = \hat{f}$.
• We denote by $S_{t_0}^{t_1}(t_2,\ell)$ the intégral $S_{t_0}^{t_1}(t_2,\ell) = \int_{t_0}^{t_1} e^{- i \ell (t_2 - s)/\varepsilon} ds = ( i \varepsilon / \ell) ( e^{- i \ell (t_2 - t_1)/\varepsilon}-e^{- i \ell (t_2 - t_0)/\varepsilon})$

Here are the calculations

• $u_{1,\ell} = \hat{U}_{n, \ell}$

• $u_{2,\ell} = e^{- i \ell h_n /(2 \varepsilon)}\hat{U}_{n, \ell} + S_0^{h_n /2} ( h_n /2,\ell ) G_{\ell}(u_1)$

• $u_{3,\ell} = e^{- i \ell h_n /(2 \varepsilon)}\hat{U}_{n, \ell} + S_0^{h_n /2} ( h_n /2,\ell ) G_{\ell}(u_2)$

• $u_{4,\ell} = e^{- i \ell h_n /(2\varepsilon)}u_{2,\ell} + S_0^{h_n/2} ( h_n/2,\ell )[ 2 G_{\ell}(u_3)-G_{\ell}(u_1)]$ (see (28) of S.M. Cox, P.C. Matthews (2002), with $c=-i \ell h_n /\varepsilon$)

From (29) of S.M. Cox, P.C. Matthews (2002), with $c=-i \ell h_n /\varepsilon$, we have

\[\hat{U}_{n+1, \ell} = e^{- i \ell h_n /\varepsilon}\hat{U}_{n, \ell} + G_{\ell}(u_1) [-4+i \ell h_n /\varepsilon + e^{-i \ell h_n /\varepsilon}(4+3i \ell h_n /\varepsilon+(i \ell h_n /\varepsilon)^2]\\+ (2 G_{\ell}(u_2) + G_{\ell}(u_3) )[-2-i \ell h_n /\varepsilon+e^{-i \ell h_n /\varepsilon}(2-i \ell h_n /\varepsilon)]\\ + G_{\ell}(u_4)[-4+3i \ell h_n /\varepsilon -(i \ell h_n /\varepsilon)^2 + e^{-i \ell h_n /\varepsilon}(4+i \ell h_n /\varepsilon)]/(h_n^2 (i \ell h_n /\varepsilon)^3)\]