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RCMS: Right correction Magnus series approach for oscillatory ODEs. (English) Zbl 1093.65071

Summary: We consider the right correction Magnus series (RCMS), a method for integrating ordinary differential equations (ODEs) of the form \(y^{\prime }=[\lambda A+A_{1}(t)]y\) 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 Frénet-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.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Software:

RCMS; SLCPM12; ode45; SLEDGE
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References:

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