THE LANCZOS OPTIMIZATION OF A SPLITTING-UP METHOD TO SOLVE HOMOGENEOUS EVOLUTIONARY EQUATIONS

Let the Cauchy problem for a symmetrical homogeneous ODE system be solved by a difference scheme and let s be the required number of matrix-vector operations with the finite-difference matrix. In classical schemes s is proportional to the number of time steps. The Lanczos method is used to decrease s without essential increase of error. A theoretical estimate is given which shows approximately the square-root s advantage of such an approach. Its application to the 2D heat conduction equation is considered. One- and two-cyclic alternating direction difference schemes are used. Some numerical experiments show that the arithmetical costs are reduced by a factor 3 up to 60 with respect to the classical approach. Combination of a splitting scheme and the Lanczos method is also proposed for the computation of the lower part of the spectrum and for solving some other problems.