[Paper Review] Analysis and Computation of a Discrete KdV-Burgers Type Equation with Fast Dispersion and Slow Diffusion

Zvi Artstein, C. W. Gear+3 more|ArXiv.org|Aug 19, 2009

Numerical methods for differential equations11 references4 citations

TL;DR

This paper develops a novel computational framework for analyzing the long-time behavior of a discrete KdV-Burgers equation with fast dispersion and slow diffusion, using invariant measures and averaging of Lax invariants to derive effective slow dynamics without requiring explicit angle variables. The method enables efficient projective integration with speedups exceeding 40:1 over direct simulation by leveraging the fast system's invariants and Young measure averaging.

ABSTRACT

The long time behavior of the dynamics of a fast-slow system of ordinary differential equations is examined. The system is derived from a spatial discretization of a Korteweg-de Vries-Burgers type equation, with fast dispersion and slow diffusion. The discretization is based on a model developed by Goodman and Lax, that is composed of a fast system drifted by a slow forcing term. A natural split to fast and slow state variables is, however, not available. Our approach views the limit behavior as an invariant measure of the fast motion drifted by the slow component, where the known constants of motion of the fast system are employed as slowly evolving observables; averaging equations for the latter lead to computation of characteristic features of the motion. Such computations are presented in the paper.

Motivation & Objective

• To analyze the long-time behavior of a discrete KdV-Burgers-type equation with fast dispersion and slow diffusion, where classical averaging methods fail due to lack of explicit angle variables.
• To develop a computational framework that bypasses the need for explicit fast-slow component splitting or knowledge of periodic orbits in the fast dynamics.
• To utilize Lax invariants and invariant measures of the fast system as slow observables to derive effective equations governing the slow diffusion process.
• To demonstrate that projective integration based on these slow observables enables significant computational speedups—over 40:1—compared to direct numerical integration.
• To validate the method using both Young measure averaging and the equation-free approach, showing indistinguishable results from direct simulations.

Proposed method

• The system is modeled as a semi-discrete approximation of a KdV-Burgers PDE, with fast dispersion from the nonlinear advection term and slow diffusion from the viscous term.
• The fast dynamics are analyzed via a Lax pair representation, which provides a rich family of first integrals (Lax invariants) that serve as slow observables.
• Averaging is performed not on the state variables but on the time-averaged values of these invariants, using invariant measures of the fast flow to compute effective drifts.
• Young measure theory is employed to rigorously justify the averaging process, where the limit measure captures the statistical behavior of the fast oscillations.
• Projective integration is applied to the slow observables: the time derivative of each invariant is approximated via numerical differencing over one fast period, then advanced in time with large steps.
• The equation-free method is implemented by simulating the full system over one fast period, computing the slope of the slow observables, and projecting forward in time without explicit knowledge of the slow equation.

Experimental results

Research questions

• RQ1 How can effective dynamics be derived for a discrete KdV-Burgers system with fast dispersion and slow diffusion when classical averaging fails due to lack of explicit angle variables?
• RQ2 Can Lax invariants of the fast system serve as reliable slow observables for constructing an effective equation governing long-time behavior?
• RQ3 To what extent can projective integration based on invariant measures and numerical differencing of observables outperform direct numerical integration in terms of computational speed?
• RQ4 How accurately do the Young measure averaging and equation-free methods reproduce the true slow dynamics of the system?
• RQ5 What is the quantitative speedup achievable by using invariant-based projective integration compared to direct simulation?

Key findings

• The method achieves a computational speedup of over 40:1 when the slow diffusion parameter ν is reduced by a factor of 10, compared to direct integration.
• Projective integration using the equation-free approach with one-period time derivatives produced results indistinguishable from direct simulation and Young measure averaging at plot accuracy.
• For ν = 0.001, tracking the decay of slow observables over a factor of 10−5 reduction required approximately 23,000 integration steps with direct simulation, but only ~5,000 steps with projective integration.
• The time derivative of the slow observables was approximated via numerical differencing over one fast period, with errors scaled by 10³ at t = 1,492.08 being less than 5.9 for v₃ when using 12 projective steps.
• The invariant measures of the fast system (3.1) were found to be supported on tori, and their decay under slow diffusion was visualized in Figure 7, confirming the slow evolution of the system's statistical structure.
• The Lax invariants v₁ to v_{N/2+1} were successfully used as slow observables, and their evolution under slow diffusion was accurately captured by both averaging and equation-free methods.