A Krylov projection method for systems of ODEs (Q1360557)

The authors describe how the Krylov projection method may be applied to a system of \(m\) ordinary linear differential equations (ODEs) (i) \(y'-Ay= f(t)v\), \(0\leq t\leq T\), \(f(t)\in\mathbb{R}\), \(y,v\in\mathbb{R}^m\), (ii) \(y(0)=0\), to obtain a system of \(k\) equations of the same form \(k\leq m\). A bound is established for the norm of the difference between the solutions of the two systems, \(r_k(t)= y_m(t)- y_k(t)\). The residual \(r_k(t)\) again satisfies an equation of the type (i), (ii). The projection method may be repeated with different \(k\) until a sufficiently small residual is obtained. Equations of the type (i), (ii) ocour in approximation to solutions of linear parabolic partial differential equations in which the space variable is discretized. Results of computations are reported in which the method was applied to two parabolic equations with known solutions. For the first equation \(m=30\), \(k=1\), \(k=10\), were used and for the second \(m=10\), \(k=1\), \(k=5\). Using repetition, residuals with norm less than \(10^{-10}\) were obtained.