Effective equations for discrete systems: A time stepper based approach

TL;DR

This paper introduces a computer-assisted method to analyze the effective continuum behavior of spatially discrete evolution equations without explicitly deriving the coarse model, utilizing time-integration and bifurcation techniques.

Contribution

The paper presents a novel approach that employs a time-stepper based method combined with bifurcation analysis to study discrete systems' effective equations without explicit model derivation.

Findings

Successfully monitors pinning of coherent structures in discrete systems.

Demonstrates the method's effectiveness through comparison with quasi-continuum approaches.

Provides a new computational framework for analyzing discrete systems' continuum limits.

Abstract

We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation (Shroff and Keller, 1993). The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Pade approximations.