Stability of the Richardson Extrapolation applied together with the [formula omitted]-method

Abstract

Consider a system of ordinary differential equations (ODEs) d y / d t = f ( t , y ) where (a) t ∈ [ a , b ] with b > a , (b) y is a vector containing s components and (c) y ( a ) is given. The θ -method is applied to solve approximately the system of ODEs on a set of prescribed grid points. If N is the number of time steps that are to be carried out, then this numerical method can be defined using the following set of relationships: y n = y n − 1 + h ( 1 − θ ) f ( t n − 1 , y n − 1 ) + h θ f ( t n , y n ) , θ ∈ [ 0.5 , 1.0 ] , n = 1 , 2 , … , N , h = ( b − a ) / N , t n = t n − 1 + h = t 0 + n h , t 0 = a , t N = b . As a rule, the accuracy of approximations { y n ∣ n = 1 , 2 , … , N } can be improved by applying the Richardson Extrapolation under the assumption that the stability of the computational process is preserved. Therefore, it is natural to require that the combined numerical method (Richardson Extrapolation + the θ -method) is in some sense stable. It is proved in this paper that the combined method is strongly A-stable when θ ∈ [ 2 / 3 , 1.0 ] . It is furthermore shown that some theorems proved in a previous paper by the same authors, Faragó et al. (2009) [1], are simple corollaries of the main result obtained in the present work. The usefulness of the main result in the solution of many problems arising in different scientific and engineering areas is demonstrated by performing a series of tests with an extremely badly scaled and very stiff atmospheric chemistry scheme which is actually used in several well-known large-scale air pollution models.