Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations

Abstract

In this paper, two multiscale time integrators (MTIs), motivated from two types of multiscale decomposition by either frequency or frequency and amplitude, are proposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter 0 < ε ≤ 1. In fact, the solution to this equation propagates waves with wavelength at O(ε 2 ) when 0 < ε ≪ 1, which brings significantly numerical burdens in practical computation. We rigorously establish two independent error bounds for the two MTIs at O(τ 2 /ε 2 ) and O(ε 2 ) for ε ∈ (0, 1] with τ > 0 as step size, which imply that the two MTIs converge uniformly with linear convergence rate at O(τ) for ε ∈ (0, 1] and optimally with quadratic convergence rate at O(τ 2 ) in the regimes when either ε = O(1) or 0 < ε ≤ τ. Thus the meshing strategy requirement (or ε-scalability) of the two MTIs is τ = O(1) for 0 < ε ≪ 1, which is significantly improved from τ = O(ε 3 ) and τ = O(ε 2 ) requested by finite difference methods and exponential wave integrators to the equation, respectively. Extensive numerical tests and comparisons with those classical numerical integrators are reported, which gear towards better understanding on the convergence and resolution properties of the two MTIs. In addition, numerical results support the two error bounds very well.

A Uniformly Accurate Multiscale Time Integrator Pseudospectral Method for the Klein--Gordon Equation in the Nonrelativistic Limit Regime

“…Downloaded 11/18/14 to 130.238.7.43. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Table 1 Spatial error analysis: e τ,h ε (T = 1) with τ = 5 × 10 −6 for different ε and h. e τ,h ε (T ) h 0 = 1 h 0 /2 h 0 /4 h 0 /8 ε 0 = 0.5 1.65E -1 3.60E -3 1.03E -6 7.34E -11 ε 0 /2 1 2.65E -1 9.70E -3 9.07E -7 5.03E -11 ε 0 /2 2 9.02E -1 1.34E -2 1.73E -7 4.60E -11 ε 0 /2 3 1.13E+0 2.98E -2 2.25E -7 4.10E -11 ε 0 /2 4 4.67E -1 3.14E -2 1.79E -7 4.78E -11 ε 0 /2 5 7.41E -1 2.73E -2 2.50E -7 5.49E -11 ε 0 /2 7 7.41E -1 2.62E -2 2.12E -7 4.96E -11 ε 0 /2 9 6.33E -1 3.57E -2 1.92E -7 5.04E -11 ε 0 /2 11 9.19E -1 2.44E -2 2.19E -7 6.18E -11 ε 0 /2 13 1.18E+0 2.38E -2 2.59E -7 5.86E -11 …”

“…If the cubic nonlinearity in the KG equation (1.1) is replaced by a general gauge invariant nonlinearity, the general MTI-FP method can be designed similar to those in [7].…”

“…On the other hand, the amplitude of z n ± is usually at O(1) and the amplitude of r n is at O(ε 2 ) = o(1) when ε is small; thus it can also be regarded as large-small amplitude decomposition. Specifically, for the pure power nonlinearity, i.e., f satisfies (1.3), explicit formulas for f ± and f r have been given in [7].…”

“…The two MTIs converge uniformly for ε ∈ (0, 1] and have some advantages compared to the FDTD and EWI as well as AP methods in integrating highly oscillatory second-order ODEs for ε ∈ (0, 1] [7], especially when ε is not too big and too small, i.e., in the intermediate regime. The aim of this paper is to Downloaded 11/18/14 to 130.238.7.43.…”

Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

“…Tabs. [13][14][15]. All the five UA methods have some temporal convergence order reductions in the resonance regime.…”