Reinhard Frank

The Concept of B-Convergence

A useful concept of convergence for nonlinear stiff ODE’s will be developed, which permits the derivation of uniform global error bounds independent of the stiffness of the considered problem. This concept will be discussed for three simple methods for stiff systems, the implicit Euler scheme, the implicit midpoint rule and the implicit trapezoidal rule.

Stability Properties of Implicit Runge–Kutta Methods

New stability concepts—

ITERATED DEFECT CORRECTION FOR THE EFFICIENT SOLUTION OF STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

BS

Iterated defect correction for differential equations part I: theoretical results

-stability and

Asymptotic error expansions for stiff equations: an analysis for the implicit midpoint and trapezoidal rules in the strongly stiff case

BSI

An extension of the applicability of iterated deffered corrections

-stability (internal

Asymptotic error expansions for stiff equations: the implicit Euler scheme

BS

The application of iterated defect correction to variational methods for elliptic boundary value problems

-stability)—are introduced.

A note on convergence concepts for stiff problems

BS

Modern convergence theory for stiff initial-value problems

-stability is a modification and extension of B-stability and enables the derivation of order results for Runge–Kutta methods applied to general nonlinear stiff initial value problems. These order results (B-consistency, B-convergence) are not affected by stiffness. Several classes of implicit Runge–Kutta methods are shown to be