Phase-Lag Analysis of Implicit Runge–Kutta Methods

Abstract

Abstract. We analyse the phase errors introduced by implicit Runge-Kuttu methods when a linear inhomogeneous test equatiotl is integrated. It is shown that the homogeneous phase errors dominate if long interval integrations are performed. Homogeneous dispersion relations for the special class of DIRK methods are derived and a few high-order dispersive DIRK methods are constructed. These methods are applied to systems of linear ditferential equations with oscillating solutions and compared with the "conventional" DIRK methods of NiHsett and Crouzeix.Key words. numerical analysis, ordinary differential equatiotlS, Runge-Kutta methods, periodic solutions AMS(MOS) subject classification. 65L05 l. Introduction. In this paper, special diagonally implicit Runge-Kutta (DIRK) methods will be constructed for integrating systems of OD Es of the formwhen we know in advance that the solution is oscillating. Analogously to a generally adopted approach in the phase-lag analysis of numerical methods for second-order equations with oscillating solutions, we use the equation (cf.as a test equation. Here w represents a natural (or eigen) frequency of the system and w,, represents the frequency of the forced solution component.In § 2, we start by deriving explicit expressions for the phase lag introduced by general, implicit Runge-Kutta (RK) methods. The phase lag is composed of two parts: the homogeneous phase lag corresponding to the eigenmodes in the solution, and the inhomogeneous phase lag corresponding to the forced solution component. We will show that in calculations over long intervals of integration, the homogeneous phase lag tends to increase linearly, whereas the inhomogeneous phase error is constant. For this reason, we concentrate on the reduction of homogeneous phase errors.In § 3, we introduce the concept of a qth order dispersive stability fimction, and we show that such a stability function generates Runge-Kutta methods that have homogeneous phase errors of order q.From § 4 on, we confine our considerations to DIRK methods. We first derive the (dispersion) relations specifying a qth order dispersive stability function (we remark that for explicit Runge-Kutta methods these relations can be found in [8]). It is shown that there exists a one-parameter family of m-stage, pth order consistent DIRK methods with homogeneous phase errors of order q=2(m-lp/2j). In § 5, the dispersion relations are solved for one-, two-, three-, and four-stage DIRK methods and the resulting stability functions are constructed. These functions are dispersive of order q = 2m. The two-stage stability function turns out to be identical with the stability function of the well-known DIRK method of NQlrsett (10].

Diagonally implicit symplectic Runge-Kutta methods with special properties

“…The phase-lag (or dispersion) property was introduced by Brusa and Nigro (1980) [1] and was extended to RK(N) methods by Van der Houwen and Sommeijer in their works [3] [4]. Based on the reasons fully described in Van der Houwen and Sommeijer [4] we shall confine our cosiderations to homogeneous phase-lag.…”

Comparison of some special optimized fourth-order Runge–Kutta methods for the numerical solution of the Schrödinger equation

“…A pioneer paper is due to Bettis [16], in which adapted RK algorithms for the integration of these problems are presented. In the mid-eighties Van der Houwen et al [17][18][19] constructed RK(N) methods with a reduced phase error. Since then, many authors have investigated the idea of minimizing the phase-lag for RK(N) methods (see for example [20]).…”

Análise das propriedades phase-lag e erro de amplificação do método RK-Butcher

“…Usando o mesmo raciocínio, analisamos, em seguida, as propriedades do método RK-Butcher de ordem algébrica 5 dado por (8) e (10). Aplicando este métodoà equação teste (1) obtemos…”

“…Aplicações dos métodos de Runge-Kutta (RK) para solução de problemas do tipo (1) foram primeiramente considerados por Brusa e Nigro [1] que introduziram a propriedade phaselag como uma ferramenta para analisar o comportamento dos métodos para problemas com soluções oscilatórias. Desde então, vários trabalhos voltados para análise e desenvolvimento de métodos do tipo explícito e implícito para equações diferencias ordinárias de primeira e segunda ordem, foram publicados, entre os quais citamos Van der Houwen e Sommeijer [9], [10], Calvo e outros [3], Chawla e Al-Zanaid [4], Senu e outros [6].…”