Computation of optimal linear strong stability preserving methods via adaptive spectral transformations of Poisson-Charlier measures

Abstract

Strong stability preserving (SSP) coefficients govern the maximally allowable step-size at which positivity or contractivity preservation of integration methods for initial value problems is guaranteed. In this paper, we show that the task of computing linear SSP coefficients of explicit one-step methods is, to a certain extent, equivalent to the problem of characterizing positive quadratures with integer nodes with respect to Poisson-Charlier measures. Using this equivalence, we provide sharp upper and lower bounds for the optimal linear SSP coefficients in terms of the zeros of generalized Laguerre orthogonal polynomials. This in particular provides us with a sharp upper bound for the optimal SSP coefficients of explicit Runge-Kutta methods. Also based on this equivalence, we propose a highly efficient and stable algorithm for computing these coefficients, and their associated optimal linear SSP methods, based on adaptive spectral transformations of Poisson-Charlier measures. The algorithm possesses the remarkable property that its complexity depends only on the order of the method and thus is independent of the number of stages. Our results are achieved by adapting and extending an ingenious technique by Bernstein in his seminal work on absolutely monotonic functions. Moreover, the techniques introduced in this work can be adapted to solve the integer quadrature problem for any positive discrete multi-parametric measure supported on $\mathbb{N}$ under some mild conditions on the zeros of the associated orthogonal polynomials.