The generalized Euler process for exponentially dominant systems
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.
scientific article; zbMATH DE number 2075449