The generalized Euler process for exponentially dominant systems (Q1433097)

Generalized Euler and Runge-Kutta algorithms are obtained for a system of exponentially dominant ordinary differential equations \(x'(t)=f(x(t))\), \(t\geq 0\), \(x(t)\in \mathbb R^n\), \(x(0) = x_n\in \mathbb R^n\) by transforming the system by change of dependent variable and applying the algorithms to the transformed system. Let the Jacobian \(J = d/dxf(0)\) be reduced to the real canonical form \(P^{-1}JP =S_0+N_0\), \(S_0\) semisimple and \(N_0\) nilpotent and let \(S=\alpha I+1/2(S_0 - S_0^T)\) where \(\alpha\) is the maximum of the real parts of the eigenvalues of \(J\). The system is transformed to a system in \(z(t) = \exp(- tS)P^{-1}x(t)\). If \(S\neq 0\) the system is exponentially dominant. Results of trials of the new method are compared to results obtained by standard methods, showing the new method to be effective.