Stepsize Conditions for Boundedness in Numerical Initial Value Problems

Abstract

Abstract. For Runge-Kutta methods (RKMs), linear multistep methods (LMMs), and classes of general linear methods (GLMs), much attention has been paid, in the literature, to special nonlinear stability requirements indicated by the terms total-variation-diminishing, strong stability preserving, and monotonicity. Stepsize conditions, guaranteeing these properties, were derived by Shu & Osher [J. Comput. Phys., 77 (1988), pp. 439-471] and in numerous subsequent papers. These special stability requirements imply essential boundedness properties for the numerical methods, among which the property of being total-variation-bounded. Unfortunately, for many well-known methods, the above special requirements are violated, so that one cannot conclude in this way that the methods are (total-variation-)bounded. In this paper, we focus on stepsize conditions for boundedness directly, rather than via the detour of the above special stability properties. We present a generic framework for deriving best possible stepsize conditions which guarantee boundedness of actual RKMs, LMMs, and GLMs, thereby generalizing results on the special stability properties mentioned above.

Special boundedness properties in numerical initial value problems

“…The following theorem is a variant of a result given earlier in the literature, see [13]. In fact, when all µ i are equal to each other, part (I) of the theorem is an immediate corollary to Theorem 2.2 in the paper just mentioned.…”

“…The necessity of the conditions (2.13) and (2.14) can be proved by almost the same arguments as already given in [13,Section 4.2]. 2 Theorem 2.5 has a wider scope, certainly, than the theorems of Section 2.2, in that µ i and µ ij are arbitrary coefficients and assumption (1.5) is not needed.…”

“…Furthermore, the parameter matrices S = (s ij ) ∈ R m×l , T = (t ij ) ∈ R m×m , corresponding to (1.2), are determined by the number of steps N as well as by the coefficients of the given GLM. Detailed examples of such representations, as well as alternative representations of actual multistep-multistage methods, can be found in Spijker [22] for N = 1 and in Hundsdorfer, Mozartova & Spijker [13] for N > 1; cf. also Section 4 of the present paper.…”

“…Stepsize conditions corresponding to general bounds (1.8) were derived, e.g., in Ruuth & Hundsdorfer [19], Hundsdorfer, Mozartova & Spijker [13]. The general bounds obtained thus far in the literature are relevant in cases where (1.7) is violated or even (1.5) is not in force.…”

“…Moreover, the boundedness theory, as given e.g. in [13] would lead to very complicated conditions. In Section 4.2 we study the two-step Adams-Bashforth method.…”