A posteriori error estimation for Magnus-type integrators

Abstract

We study high-order Magnus-type exponential integrators for large systems of ordinary differential equations defined by a time-dependent skew-Hermitian matrix. We construct and analyze defect-based local error estimators as the basis for adaptive stepsize selection. The resulting procedures provide a posteriori information on the local error and hence enable the accurate, efficient, and reliable time integration of the model equations. The theoretical results are illustrated on two numerical examples .

Symmetrized local error estimators for time-reversible one-step methods in nonlinear evolution equations

“…Various versions of the resulting classical defect-based error estimators for these exponential integrators are presented in [12]. We now follow two of these approaches.…”

“…We now follow two of these approaches. To keep the presentation self-contained within reason, we briefly recapitulate the underlying material from [12,Section 3], and we introduce the corresponding symmetrized defect approximations.…”

“…One possible computable approximation is obtained by truncating the series (5.18); we will refer to the resulting procedure as Taylor variant. The procedure in conjunction with the classical defect is given in detail in [12,Section 3]. We remark at this point that symmetry of the basic CFM integrator implies that truncation of the series (5.18) at m = p, i.e., approximating Γ j (τ, t 0 ) by 12…”

“…We will refer to the resulting procedure as Hermite variant. The procedure in conjunction with the classical defect was also introduced in [12,Section 3]. Similarly as for the Taylor variant, it can be shown that quadrature of order p is sufficient to obtain a defect approximation of order p + 2.…”

“…As an example we consider the classical fourth order Magnus integrator based on quadrature at Gaussian points (see [12]), S(τ, t 0 ) = e Ω(τ,t0) = e τ B(τ,t0) , (5.20a) where Ω(τ, t 0 ) = τ B(τ, t 0 ) approximates the Magnus series Ω(τ, t 0 ),…”