Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics

TL;DR

This paper introduces a diffusive approximation method for a time-fractional Burger's equation in nonlinear acoustics, enabling efficient numerical modeling of wave attenuation due to fractional derivatives.

Contribution

It develops a diffusive representation and an optimized quadrature approach for local-in-time numerical simulation of fractional derivatives in nonlinear wave equations.

Findings

Efficient numerical scheme for fractional Burger's equation

Demonstrates the impact of fractional attenuation on wave propagation

Provides a stable and accurate computational method

Abstract

A fractional time derivative is introduced into the Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling, and to illustrate the effect of the fractional attenuation on the wave propagation.