On application of the Lanczos method to solution of some partial differential equations

Abstract

Let A be a square symmetric n × n matrix, φ be a vector from R n , and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f( A) φ arises very often in mathematical physics. We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos decomposition method (SLDM) solution as u m = ∥ φ ∥ Qf( H) e 1, where Q is the n × m matrix of the m Lanczos vectors and H is the m × m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for ∥ u − u m ∥ that are stable in the presence of computer round-off errors when using the simple Lanczos method. We concentrate on computation of exp(− tA)φ, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations.