Positivity of an explicit Runge–Kutta method

Abstract

This paper deals with the numerical solution of initial value problems (IVPs), for systems of ordinary differential equations (ODEs), by an explicit fourth-order Runge-Kutta method (we will refer to it as the classical fourth-order method) with special nonlinear stability property indicated by the positivity. Stepsize conditions, guaranteeing this property based on general theory, have been studied earlier, see e.g. Hundsdorfer and Verwer (2003). In this paper we show that general obtained result on positivity for classical fourth-order method is somewhat too strict. We obtain new results for positivity which are important in practical applications. We provide some computational experiments to illustrate our results. Ó 2015 Faculty of Engineering, Ain Shams University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Positivity for Convective Semi-discretizations

“…We thank Zoltán Horváth (Széchenyi István University, Hungary) for pointing this out 3. Our Proposition 8 seems to directly contradict Theorem 1 in[13]. To explain the discrepancy, note that the polynomial P 3 in our proof becomes negative along a 9-dimensional hyperface of the hypercube [0, ε] 10 for any ε > 0; in[13] it seems that the non-negativity of the corresponding (but slightly different) polynomial was checked only at the vertices of the hypercube [0, 1] 10 .…”

“…Given an ERK method, it is natural to consider the factorization ∆t ≤ ∆t 0 ≡ γτ 0 (11) of the maximum allowed step size ∆t 0 , with τ 0 appearing in Example 1 or 2, because (i) this product structure naturally arises during the computations (see (13)), and (ii) the factor γ, referred to as the (positivity) step-size coefficient, will depend only on the chosen ERK method, whereas τ 0 depends on the right-hand side of the problem (10). We will sometimes write γ(A, b), where (A, b) refer to the Butcher coefficients of the RK method, to emphasize that the step-size coefficient depends on the method coefficients.…”

“…In this section we generalize and make systematic an approach that was introduced in [12,13]. Applying an explicit RK method to (10) we obtain…”

“…Definition 1. Let an explicit Runge-Kutta method be given with coefficients A, b, and let P i denote polynomials defined implicitly by (14) when the method is applied to (10) and ξ is defined by (13). The (positivity) step-size coefficient of the method is…”

“…Examples of such PDEs include scalar hyperbolic conservation laws in one spatial dimension, as well as the heat equation and some of its generalizations. In this work we study a technique that was first used in [12,13] for determining whether a certain class of numerical discretizations satisfies the discrete analog of (1) or (2). We focus on the application of this technique to the initial-boundary-value problem given by the hyperbolic conservation law (6) below, together with positive initial and boundary data.…”