Computational coarse graining of a randomly forced 1-D Burgers equation

TL;DR

This paper presents a computational method for coarse graining the large-scale dynamics of a randomly forced 1D Burgers equation, enabling efficient extraction of macroscopic features without explicit derivation of effective equations.

Contribution

It introduces coarse projective integration and dynamic renormalization techniques to analyze the energy spectrum evolution in the Burgers equation.

Findings

Demonstrates self-similar energy spectrum evolution over time.

Shows acceleration of macroscopic analysis using short simulation bursts.

Extracts dynamic exponents without explicit effective equations.

Abstract

We explore a computational approach to coarse graining the evolution of the large-scale features of a randomly forced Burgers equation in one spatial dimension. The long term evolution of the solution energy spectrum appears self-similar in time. We demonstrate coarse projective integration and coarse dynamic renormalization as tools that accelerate the extraction of macroscopic information (integration in time, self-similar shapes, and nontrivial dynamic exponents) from short bursts of appropriately initialized direct simulation. These procedures solve numerically an effective evolution equation for the energy spectrum without ever deriving this equation in closed form.