[Paper Review] Nonlinear Geometric Optics method based multi-scale numerical schemes for highly-oscillatory transport equations

Nicolas Crouseilles, Shi Jin+1 more|arXiv (Cornell University)|May 31, 2016

Advanced Numerical Methods in Computational Mathematics23 references3 citations

TL;DR

This paper introduces a novel asymptotic-preserving numerical scheme for highly oscillatory transport equations by embedding the oscillatory phase into an independent variable via a nonlinear geometric optics ansatz. The method achieves uniform first-order accuracy in the wave period without resolving high-frequency oscillations, enabling stable and accurate simulations with mesh and time step independent of the small parameter ε, as validated by numerical tests across multiple scales.

ABSTRACT

We introduce a new numerical strategy to solve a class of oscillatory transport PDE models which is able to captureaccurately the solutions without numerically resolving the high frequency oscillations {\em in both space and time}.Such PDE models arise in semiclassical modeling of quantum dynamics with band-crossings, and otherhighly oscillatory waves. Our first main idea is to use the nonlinear geometric optics ansatz, which builds theoscillatory phase into an independent variable. We then choose suitable initial data, based on the Chapman-Enskog expansion, for the new model. For a scalar model, we prove that so constructed model will have certain smoothness, and consequently, for a first order approximation scheme we prove uniform error estimates independent of the (possibly small) wave length. The method is extended to systems arising from a semiclassical model for surface hopping, a non-adiabatic quantum dynamic phenomenon. Numerous numerical examples demonstrate that the method has the desired properties.

Motivation & Objective

• To develop a numerical method that accurately captures solutions of highly oscillatory transport equations without resolving the small wave length ε.
• To overcome the computational infeasibility of standard schemes that require Δx, Δt = O(ε) for high-frequency oscillations.
• To extend asymptotic-preserving (AP) schemes to spatially and temporally oscillatory problems by incorporating geometric optics principles.
• To ensure uniform error estimates independent of ε through a carefully constructed initial condition based on Chapman-Enskog expansion.
• To apply the method to systems arising in semiclassical surface hopping models involving non-adiabatic quantum transitions.

Proposed method

• The method employs a nonlinear geometric optics (NGO) ansatz to transform the oscillatory phase into an independent variable τ, decoupling high-frequency oscillations from physical space and time.
• A new system of equations is derived for the amplitude function V, which evolves under a modified transport equation with phase-dependent coefficients.
• Initial data are constructed using a Chapman-Enskog-type expansion to ensure smoothness and consistency with the original problem’s oscillatory structure.
• A time-splitting spectral method is used for spatial discretization and exact time integration for transport and relaxation terms, while the fast oscillation in τ is handled via pseudo-spectral methods.
• For systems, the method extends to 2×2 systems modeling surface hopping, with a transformation to a system in (x, p, τ) variables and exact integration in τ.
• The scheme avoids resolving ε by using a coarse mesh (Δx ≫ ε) and fixed time step, while maintaining accuracy through phase variable transformation and proper initial data.

Experimental results

Research questions

• RQ1 Can a numerical scheme be designed to capture highly oscillatory solutions of transport equations without resolving the small wave length ε?
• RQ2 Does the nonlinear geometric optics ansatz enable uniform convergence of the numerical solution in the limit ε → 0?
• RQ3 Can the Chapman-Enskog expansion be used to construct initial data that preserve smoothness and accuracy in the transformed system?
• RQ4 How does the method perform for systems with non-adiabatic transitions, such as in surface hopping models?
• RQ5 Can the method maintain accuracy and efficiency when ε is reduced to 1/256, where standard schemes become computationally prohibitive?

Key findings

• The method achieves uniform first-order error estimates in the wave period for the scalar model, independent of ε, proving convergence without requiring Δx or Δt to scale with ε.
• For ε = 1, the new method with N_x = 32, N_p = 64, N_τ = 8 matches the reference solution (Δt = 0.05, N_x = 256) in accuracy, with CPU time reduced from 15 s to 1 min.
• For ε = 1/32, the new method maintains accuracy with the same coarse mesh (N_x = 32), while the reference method required Δt = 0.02 and N_x = 512, with CPU time increasing to 75 s.
• For ε = 1/256, the new method still produces accurate results with fixed parameters (Δt = 0.05, N_x = 32), while the reference method required Δt = 0.0005 and N_x = 4096, with CPU time of 1420 s.
• Numerical results show that the method captures both the point-wise solution and densities accurately, even when the mesh is coarser than the spatial oscillations.
• The method successfully handles complex systems such as the 2×2 surface hopping model, demonstrating robustness and scalability across multiple orders of magnitude in ε.