On the error structure of the implicit Euler scheme applied to stiff systems of differential equations
This paper is concerned with the existence and behavior of asymptotic expansions of the global error when the implicit Euler scheme is applied to a linear, self-adjoint constant coefficient stiff system of differential equations with the stiff eigenvalues arbitrarily distributed on the negative real axis. It must be pointed out that in earlier papers on this subject the existence of asymptotic expansions in the form \(e(t)=\sum^{q}_{j=p}h^ je_ j(t)+O(h^{q+1})\) has already been proved but the size of the O-constant was not studied and therefore the existence of such expansions may be completely useless in practice. Here the existence of asymptotic expansions called asymptotically correct, i.e. with moderate O-constants, is addressed. The main result is that with the above assumptions an asymptotically correct expansion does not exist immediately after the initial point due to the existence of irregular error components, but these irregular terms are damped (algebraically) along the integration interval. To confirm the above results the author studies the behavior of the error for a linear problem with a matrix arising from the standard space discretization of the one- dimensional heat equation on a uniform mesh.
scientific article; zbMATH DE number 4123192
scientific article