RCMS: Right Correction Magnus Series approach for oscillatory ODEs

Abstract

We consider RCMS, a method for integrating differential equations of the form y = [ A + A 1 (t)]y with highly oscillatory solution. It is shown analytically and numerically that RCMS can accurately integrate problems using stepsizes determined only by the characteristic scales of A 1 (t), typically much larger than the solution "wavelength". In fact, for a given t grid the error decays with, or is independent of, increasing solution oscillation. RCMS consists of two basic steps, a transformation which we call the right correction and solution of the right correction equation using a Magnus series. With suitable methods of approximating the highly oscillatory integrals appearing therein, RCMS has high order of accuracy with little computational work. Moreover, RCMS respects evolution on a Lie group. We illustrate with application to the 1D Schrödinger equation and to Frenet-Serret equations. The concept of right correction integral series schemes is suggested and right correction Neumann schemes are discussed. Asymptotic analysis for a large class of ODEs is included which gives certain numerical integrators converging to exact asymptotic behaviour.

Efficient computation of high index Sturm–Liouville eigenvalues for problems in physics

“…Thus, to compute y i = e hĀ e σ(h) y i−1 with h = h i , we need to approximate σ(h) by truncating the expansion (27) and replacing integrals by quadrature (see 4.3). As shown in [26], truncating all but the first integral leads to a fourth order method, while including also σ 2 gives us a scheme of order eight. Having approximated σ(h), its 2 × 2 matrix exponential must be computed.…”

“…When scanning over λ we have used only three values, those such that Z(h i ) = (q − λ)h 2 i = −m 2 π 2 , m = 0, 1, 2. The selection of only these was mainly intended to speed up the evaluation but this is enough as the error decreases like O(1/ √ λ) for large Z-values [26,28], and as confirmed by experimental tests showing that the error is indeed larger for smaller values of Z.…”

Uniform and high-order discretization schemes for Sturm–Liouville problems via Fer streamers

“…1-2 possess a unique countable family of solutions {λ j , y λ j } j ∈Z + 0 and that its eigenvalues are simple, bounded from below and accumulate only at infinity as well as that its eigenfunctions oscillate as functions of the eigenvalues [13], namely λ j < λ j +1 , lim j →+∞ (λ j /j 2 ) = (π/(b − a)) 2 , y λ j has exactly j zeros in (a, b).…”

“…e.g., in the piecewise perturbation methods [7], in the right-correction Magnus series [2] and in the modified Magnus methods [9]. In addition, one can consider another regime.…”

“…In addition, quadrature estimates under λ,h 1 are often valid only in the absence of critical points and subject to a non-resonance condition [6]. In the context of the integral series in [2,7,9,14] and this paper, these considerations are key since the non-resonance condition is not satisfied in the integral (which appears in higher order approximations)…”