Post by matthorr on Oct 7, 2015 20:33:49 GMT
Applying the representative ODE \( u'_n = \lambda u_n+ae^{\mu hn} \)
\[\tilde{u}_{n+a_1}=u_n+a_1h\left(\lambda u_n+ae^{\mu hn}\right)\]
\[\tilde{u}'_{n+a_1} = \lambda\Big( u_n+a_1h\left(\lambda u_n+ae^{\mu hn}\right)\Big)+ae^{\mu h(n+a_1)}\]
\[\hat{u}_{n+a_2}=u_n+a_2h\left[\lambda\left(u_n+a_1h\left(\lambda u_n+ae^{\mu hn}\right)\right)+ae^{\mu h(n+a_1)}\right]\]
\[\hat{u}'_{n+a_2} = \lambda\Bigg(u_n+a_2h\left[\lambda\left(u_n+a_1h\left(\lambda u_n+ae^{\mu hn}\right)\right)+ae^{\mu h(n+a_1)}\right] \Bigg)+ae^{\mu h(n+a_2)}\]
\[u_{n+1}=u_n+a_3h\Bigg[\lambda\Bigg(u_n+a_2h\left[\lambda\left(u_n+a_1h\left(\lambda u_n+ae^{\mu hn}\right)\right)+ae^{\mu h(n+a_1)}\right] \Bigg)+ae^{\mu h(n+a_2)}\Bigg]\]
\[u_{n+1}-\Big(1+a_3\lambda h+a_2a_3\lambda^2h^2+a_1a_2a_3\lambda^3h^3\Big)u_n = \left(a_3he^{\mu ha_2}+a_2a_3\lambda h^2e^{\mu ha_1} + a_1a_2a_3\lambda^2h^3\right) ae^{\mu hn}\]
Applying the displacement operator to the left hand side gives
\[P(E) = E-\Big(1+a_3\lambda h+a_2a_3\lambda^2h^2+a_1a_2a_3\lambda^3h^3\Big) \]
whose root is
\[ \sigma_1 = 1+a_3\lambda h+a_2a_3\lambda^2h^2+a_1a_2a_3\lambda^3h^3\]
To minimize \( er_\lambda \), the coefficients of \( \sigma \) must match those of the expansion of
\[e^{\lambda h} = 1+\lambda h + \frac{1}{2}\lambda^2h2+\frac{1}{6}\lambda^3h^3+\cdots\]
giving \( a_1 = \frac{1}{3},\ a_2 = \frac{1}{2},\ a_3 = 1 \).
\[\tilde{u}_{n+a_1}=u_n+a_1h\left(\lambda u_n+ae^{\mu hn}\right)\]
\[\tilde{u}'_{n+a_1} = \lambda\Big( u_n+a_1h\left(\lambda u_n+ae^{\mu hn}\right)\Big)+ae^{\mu h(n+a_1)}\]
\[\hat{u}_{n+a_2}=u_n+a_2h\left[\lambda\left(u_n+a_1h\left(\lambda u_n+ae^{\mu hn}\right)\right)+ae^{\mu h(n+a_1)}\right]\]
\[\hat{u}'_{n+a_2} = \lambda\Bigg(u_n+a_2h\left[\lambda\left(u_n+a_1h\left(\lambda u_n+ae^{\mu hn}\right)\right)+ae^{\mu h(n+a_1)}\right] \Bigg)+ae^{\mu h(n+a_2)}\]
\[u_{n+1}=u_n+a_3h\Bigg[\lambda\Bigg(u_n+a_2h\left[\lambda\left(u_n+a_1h\left(\lambda u_n+ae^{\mu hn}\right)\right)+ae^{\mu h(n+a_1)}\right] \Bigg)+ae^{\mu h(n+a_2)}\Bigg]\]
\[u_{n+1}-\Big(1+a_3\lambda h+a_2a_3\lambda^2h^2+a_1a_2a_3\lambda^3h^3\Big)u_n = \left(a_3he^{\mu ha_2}+a_2a_3\lambda h^2e^{\mu ha_1} + a_1a_2a_3\lambda^2h^3\right) ae^{\mu hn}\]
Applying the displacement operator to the left hand side gives
\[P(E) = E-\Big(1+a_3\lambda h+a_2a_3\lambda^2h^2+a_1a_2a_3\lambda^3h^3\Big) \]
whose root is
\[ \sigma_1 = 1+a_3\lambda h+a_2a_3\lambda^2h^2+a_1a_2a_3\lambda^3h^3\]
To minimize \( er_\lambda \), the coefficients of \( \sigma \) must match those of the expansion of
\[e^{\lambda h} = 1+\lambda h + \frac{1}{2}\lambda^2h2+\frac{1}{6}\lambda^3h^3+\cdots\]
giving \( a_1 = \frac{1}{3},\ a_2 = \frac{1}{2},\ a_3 = 1 \).