Efficient sixth order methods for nonlinear oscillation problems

Abstract

Abstract.A class of two-step (hybrid) methods is considered for solving pure oscillation second order initial value problems. The nonlinear system, which results on applying methods of this type to a nonlinear differential system, may be solved using a modified Newton iteration scheme. From this class the author has derived methods which are fourth order accurate, P-stable, require only two (new) function evaluations per iteration and have a true real perfect square iteration matrix. Now, we propose an extension to sixth order, P-stable methods which require only three (new) function evaluations per iteration and for which the iteration matrix is a true real perfect cube.This implies that at most one real matrix "must be factorised at each step. These methods have been implemented in a new variable step, local error controlling code.

Adaptive two‐step methods for semidiscretized wave equations

“…= a cos(8 + n v), t, = nh (10) where a and 8 are real constants determined by the initial conditions of (9). When a two-step method is applied to (9), the numerical solution is given by the homogeneous version of (6) obtained from ( 5 ) in the case qo = q1 = q2 = 0. After some manipulation, it can be expressed in a similar form to (10):…”

“…Based on these methods, in Section 4, we derive two-step methods for equation (1) in the non-scalar case. They are derived following the notion of diagonally implicit RK-methods (see References 8,9,11 and 17), by using efficient rational approximations to cos v, v > 0. A number of second-order and fourth-order methods with small dispersion errors are presented.…”

On the generation of mono-implicit Runge-Kutta-Nyström methods by mono-implicit Runge-Kutta methods

“…In [10] it has been shown that a one-parameter family of double-stride L-stable methods of fourth-order, obtained by the coupling of three linear multistep methods [8] can also be included within the framework of MIRK methods. P-stable methods for second-order ODEs of the form y"= f(t, y) are constructed either by considering so-called symmetric hybrid methods [3,4,9,[17][18][19] or by using IRKN methods [12]. In the two previous papers [23,24] the present authors have studied a special class of four-stage fourthorder P-stable MIRKN methods, which are characterized by stages evaluated at abscissas having integer values in the range 0-3.…”

Υπολογιστικές Μέθοδοι Για Την Αριθμητική Επίλυση Συνήθων Διαφορικών Εξισώσεων

“…al [132] -[136],. Voss και Serbin[137],Thomas [138] -[141] και Coleman[142].Οι Brusa και Nigro[143] εισήγαγαν την ιδιότητα της υστέρησης φάσης (phase-lag), η οποία είναι µία πολύ σηµαντική ιδιότητα, την οποία πρέπει να έχουν οι µέθοδοι για την επίλυση του προβλήµατος (1.1). Σ' αυτή την ιδιότητα βασίστηκαν οι Chawla και Rao[132] και κατασκεύασαν Numerov-type µέθοδο τέταρτης αλγεβρικής τάξης και µε έκτης τάξης υστέρηση φάσης.…”