Adapted Falkner-Type Methods

Abstract

Chapter 6 establishes the multidimensional adapted Falkner-type methods for the oscillatory second-order system y″+My=f(x,y) with a symmetric positive semi-definite principal frequency matrix M∈ℝ^d×d. Adapted generating functions are formulated for deriving the coefficients of adapted Falkner-type methods. Based on the discrete Gronwall’s inequality, uniform bounds for the local truncation errors of the solution and the derivative are obtained, respectively. These error bounds turn out to be independent of the frequency matrix M. Zero-stability as well as linear stability of adapted Falkner-type methods is also analyzed. The high efficiency of adapted Falkner-type methods is illustrated by numerical examples such as the coupled oscillators, the sine-Gordon equation and a nonlinear wave equation.

Appendices

Appendix A: Derivation of Generating Functions (6.14) and (6.15)

In what follows we verify the formula (6.14) for the coefficients β _j (V).

$$ \begin{aligned} G_\beta(t,V)&=\sum _{j=0}^{\infty}\beta_j(V)t^j \\[-2pt] & =\sum_{j=0}^{\infty}(-1)^j\int _0^1(1-z)\phi_1 \bigl((1-z)^2V \bigr) \left ( \begin{array}{c}-z\\j \end{array} \right ) \,{\rm d}z\cdot t^j \\[-2pt] & =\int_0^1(1-z)\phi_1 \bigl((1-z)^2V \bigr)\sum_{j=0}^{\infty }(-t)^j \left ( \begin{array}{c}-z\\j \end{array} \right ) \,{\rm d}z \\[-2pt] & =\int_0^1(1-z)\phi_1 \bigl((1-z)^2V \bigr) (1-t)^{-z} \,{\rm d}z \\[-2pt] & =\sum_{k=0}^{\infty}\frac{(-1)^kV^k}{(2k+1)!}\int _0^1(1-z)^{2k+1}(1-t)^{-z} \,{\rm d}z. \end{aligned} $$

Integration by parts yields

$$ \int_0^1(1-z) (1-t)^{-z} \,{\rm d}z =\frac{1}{\ln(1-t)}+\frac{t}{(1-t)\ln^2(1-t)}. $$

(6.54)

For k≥1 we have

(6.55)

Based on the above analysis, we have

(6.56)

where

(6.57)

(6.58)

(6.59)

Substituting (6.57), (6.58) and (6.59) into (6.56) gives

$$ \begin{aligned}[b] G_\beta(t,V)&= \frac{1}{\ln(1-t)}\phi_1(V)+\frac{1}{(1-t)\ln^2(1-t)}I-\frac{\phi_0(V)}{\ln^2(1-t)} \\ &\quad+\frac{-V}{\ln^{2}(1-t)} G_\beta(t,V). \end{aligned} $$

(6.60)

This gives the expression for the generating functions G _β (t,V) in (6.14).

The proof of the expression for $\gamma^{*}_{j}(V)$ in (6.15) is similar.

Appendix B: Proof of (6.24)

We verify the formula (6.24) by induction on m. For m=1, the result is trivial. Assume that (6.24) holds for m=k. Then we have

$$ \begin{aligned} Q^{k+1}&=Q^kQ=\left ( \begin{array}{c@{\quad}c} \phi_{0}(k^2V) & kh\phi_{1}(k^2V)\\ -khM\phi_{1}(k^2V) & \phi_{0}(k^2V) \end{array} \right ) \left ( \begin{array}{c@{\quad}c} \phi_{0}(V) & h\phi_{1}(V)\\ -hM\phi_{1}(V) & \phi_{0}(V) \end{array} \right ) \\ &=\left ( \begin{array}{c@{\quad}c} \phi_{0} ((k+1)^2V ) & (k+1)h\phi_{1} ((k+1)^2V )\\ -(k+1)hM\phi_{1} ((k+1)^2V ) & \phi_{0} ((k+1)^2V ) \end{array} \right ). \end{aligned} $$

The last equality follows from the following equations:

$$ \begin{aligned} &c\phi_1 \bigl(c^2V\bigr)\phi_0(V)+\phi_1(V) \phi_0\bigl(c^2V\bigr)=(1+c)\phi_1 \bigl((1+c)^2V \bigr), \\ &\phi_0\bigl(c^2V\bigr)\phi_0(V)-cV \phi_1(V)\phi_1\bigl(c^2V\bigr) = \phi_0 \bigl((1+c)^2V \bigr). \end{aligned} $$

(6.61)

The first formula in (6.61) has been proved in Chap. 5 (see (5.11)). The second formula in (6.61) can be achieved by computation:

(6.62)

The material of this chapter is based on Li and Wu [8].