Accuracy and stability of multistage multistep formulas
The research concerning relations between accuracy and stability of time discretizations started by the authors [Numer. Math. 37, 61-91 (1981; Zbl 0457.65054) and ibid. 40, 245-296 (1982; Zbl 0493.65038)], is continued. Here we concentrate, rather than on the size and shape of the stability region, on the obtainable accuracy when the number of parameters in the formulas is large. By combining proofs of Reimer and Dahlquist we obtain a simpler proof for the order barrier of stable m- stage k-step formulas of \textit{M. Reimer} [SIAM J. Numer. Anal. 5(1968), 725-738 (1969; Zbl 0187.109)]. This new tool is used to give lower bounds for the error constants and to obtain fastest decay rates for the scaled error constants when either the step number k or the stage number m grows while the other stays small. Similar results are given under the stronger assumption that the formula is not only stable at the origin but that all roots except the principle one should be strictly smaller than one in modulus. The new technique is used also to give a bound for the order of formulas which are stable at infinity. Since lower bounds for the error constants are interesting only when the order has been pushed to be as high as possible we investigate the size of the Peano-kernels for the cases where formulas contain ''free'' parameters. While upper bounds for the \(L^ 1\)-norms of Peano-kernels of multistage multistep formulas are given we mainly concentrate on sharp bounds for linear multistep methods. The size of stability regions of sequences of highly accurate schemes is investigated. It is shown that if we are given a sequence of multistep multistage schemes which, in a rather mild sense become arbitrarily accurate, then either the number of stages must tend to infinity or the stability regions must shrink down.
stability regions of highly accurate families of formulas
scientific article; zbMATH DE number 3898145