On the frequency choice in trigonometrically fitted methods

Abstract

a b s t r a c tThe choice of frequency in trigonometrically fitted methods is a fundamental question, especially if long-term prediction is considered. For linear oscillators, the frequency of the method is the same as the frequency of the solution of the differential equation. However, for nonlinear problems the frequency of the method is, in general, different from the frequency of the true solution. We present some experiments showing how the frequency depends strongly on certain values.

Block third derivative method based on trigonometric polynomials for periodic initial-value problems

“…A classical procedure for estimating the frequency is not available, however, some techniques for estimating the frequency are given in [13,31,32]. A preliminary testing indicates that a good estimate of the frequency can be obtained by demanding that LT E(8) = 0, and solving for the frequency.…”

Two new embedded pairs of explicit Runge–Kutta methods adapted to the numerical solution of oscillatory problems

“…Since the analytical solutions of these IVPs are usually not available, they can be solved by using general purpose numerical methods or by using codes specially adapted to the oscillatory behavior of their solutions. The design and construction of RK methods specially adapted to the numerical solution of oscillatory IVPs (1) has been considered by several authors (see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and references therein). The coefficients of these methods usually depend on the parameter m ¼ x h, where h is the integration step-size and x represents an approximation of the main frequency of the problem.…”

Exponentially Fitted Finite Difference Schemes for Reaction-Diffusion Equations

“…For ordinary differential equations, the problem of how to choose fitting parameters are discussed in [12,13]. However, the usage EF method in the full discretization of PDE's has not been well studied.…”

“…EF solution is computed by Equations (15) and (13). Table 3 presents numerical solutions at t = 4 and x ∈ [−30, 30] with ∆x = 1, ∆t = 0.1 and for a = 1/24 , b = c = 1 .…”