Explicit Nordsieck methods with quadratic stability

Abstract

We describe the construction of explicit Nordsieck methods with s stages of order p = s−1 and stage order q = p with inherent quadratic stability and quadratic stability with large regions of absolute stability. Stability regions of these methods compare favorably with stability regions of corresponding general linear methods of the same order with inherent Runge–Kutta stability

Order conditions for general linear methods

“…Such high order and stage order methods with some desirable stability properties (large regions of absolute stability for explicit methods, A-, L-, and algebraic stability for implicit methods) were investigated in a series of papers [29,56,30,[16][17][18][19][20][21][22][23][24][25][26][27]33,34,[57][58][59] and the monograph [3]. In the next section we will derive the general order conditions for GLMs (1.5) without restrictions on stage order (except stage preconsistency and stage consistency conditions (3.11) and (3.12)) using the approach proposed by Albrecht [60][61][62][63][64] in the context of RK, composite and linear cyclic methods, and generalized by Jackiewicz and Tracogna [39,40] and Tracogna [43], to TSRK methods for ODEs, by Jackiewicz and Vermiglio [65] to general linear methods with external stages of different orders, and by Garrappa [66] to some classes of Runge-Kutta methods for Volterra integral equations with weakly singular kernels.…”

“…GLMs were first introduced by Butcher [4] using somewhat different notation, and the modern theory of these methods was developed in [5][6][7]2,8,3,9]. These methods include as special cases many numerical methods for ODEs, for example, linear multistep methods [2,8,[10][11][12][13][14], predictor-corrector methods [11][12][13][14], Runge-Kutta (RK) methods [5,6,15,2,11,12], diagonally implicit multistage integration methods (DIMSIMs) [16][17][18][19][20][21][22][23][24][25][26][27], Nordsieck methods [28][29][30][31][32][33][34], two-step Runge-Kutta (TSRK) methods [35,28,[36][37][38]…”

P-stable general Nyström methods fory″=f(y(t

“…Following the above described lines drawn in the literature in the context of GLMs for first order ODEs (also compare [28][29][30][31]), we introduce an analogous notion of stability for GLN methods (1.2), in order to let these methods inherit the same stability properties of a certain RKN method assumed as a reference.…”

Construction of Nordsieck Second Derivative General Linear Methods With Inherent Quadratic Stability

“…In this paper, we are going to relax the concept of SIRKS to the concept of inherent quadratic stability (IQS). This property was first introduced in [19] for two-step Runge-Kutta (TSRK) methods and then presented for GLMs [6,14] which guarantees the stability function has only two nonzero roots. Using this approach, we solve fewer equations in comparison with methods based on SIRKS, which makes construction to be easier and gains some additional free parameters.…”