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**Stepsize restrictions for stability of one-step methods in the numerical solution of initial value problems.**
*(English)*
Zbl 0579.65092

This important paper considers one-step discretizations \(U_{n+1}=\Phi (hA)U_ n\) of differential systems \((d/dt)u(t)=Au(t)\), with A a constant \(s\times s\) matrix and \(\Phi\) a rational function. Two numbers r, R are associated with \(\Phi\) : the stability radius r is the largest positive number such that the complex disk of radius r centered at -r is contained in the stability region of \(\Phi\) ; the contractivity radius R is the supremum of the values \(\rho\) such that \(\Phi\) and all its derivatives are nonnegative in the interval [-\(\rho\),0]. For matrices A within a suitable class, it is shown how to use r and R in the derivation of conditions on the step-size h which guarantee either a polynomial growth of \(\| U_ n\|\) as \(n\to \infty\) or a contractive behaviour, i.e. \(\| U_{n+1}\| \leq \| U_ n\|\). Of interest here is that the bounds obtained are uniform both in the dimension s and in the stiffness of A. As a consequence, the results presented are useful in the investigation of stability and convergence of one-step discretizations of evolutionary PDEs. Another attractive feature is that the norms employed do not necessarily stem from an inner product. As an example the author studies the maximum norm stability of a \(\theta\)-method, upwind scheme for the advection-diffusion problem. It should also be mentioned that the bounds presented in the paper are optimal, in the sense that a problem in the class considered exists for which step-sizes h violating those bounds originate vectors \(U_ n\) which either grow faster than polynomially or do not behave contractively.

Reviewer: J.M.Sanz-Serna

### MSC:

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

34A30 | Linear ordinary differential equations and systems |

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |