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

Spijker, M. N.

Math. Comput. 45, 377-392 (1985).

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

Keywords:

stiff systems; differential systems; stability radius; stability region; contractivity radius; convergence; upwind scheme; advection-diffusion problem