A Regularization of Burgers Equation using a Filtered Convective Velocity

TL;DR

This paper introduces a regularized version of Burgers equation using a filtered convective velocity, providing theoretical proofs, invariant analysis, and numerical simulations to understand shock behavior and solution smoothness.

Contribution

The paper presents a novel regularization method for Burgers equation with a general class of filters, including existence, uniqueness, invariants, and convergence results.

Findings

Traveling wave solutions converge to weak solutions of inviscid Burgers.

Shock behavior and thickness are characterized in simulations.

Energy spectra relate to solution smoothness.

Abstract

This paper examines the properties of a regularization of Burgers equation in one and multiple dimensions using a filtered convective velocity, which we have dubbed as convectively filtered Burgers (CFB) equation. A physical motivation behind the filtering technique is presented. An existence and uniqueness theorem for multiple dimensions and a general class of filters is proven. Multiple invariants of motion are found for the CFB equation and are compared with those found in viscous and inviscid Burgers equation. Traveling wave solutions are found for a general class of filters and are shown to converge to weak solutions of inviscid Burgers equation with the correct wave speed. Accurate numerical simulations are conducted in 1D and 2D cases where the shock behavior, shock thickness, and kinetic energy decay are examined. Energy spectrum are also examined and are shown to be related to the smoothness of the solutions.