Wavelet methods to eliminate resonances in the Galerkin-truncated Burgers and Euler equations

TL;DR

This paper introduces wavelet-based filtering techniques to eliminate resonance phenomena in Galerkin-truncated Burgers and Euler equations, improving convergence to physically relevant solutions.

Contribution

It extends wavelet filtering methods from complex-valued frames to real-valued orthogonal bases with a safety zone, effectively removing resonances in hyperbolic conservation laws.

Findings

Wavelet filtering removes resonances in Burgers equation

Method extends to 2D Euler equations

Real-valued wavelet basis with safety zone is effective

Abstract

It is well known that solutions to the Fourier-Galerkin truncation of the inviscid Burgers equation (and other hyperbolic conservation laws) do not converge to the physically relevant entropy solution after the formation of the first shock. This loss of convergence was recently studied in detail in [S. S. Ray et al., Phys. Rev. E 84, 016301 (2011)], and traced back to the appearance of a spatially localized resonance phenomenon perturbing the solution. In this work, we propose a way to remove this resonance by filtering a wavelet representation of the Galerkin-truncated equations. A method previously developed with a complex-valued wavelet frame is applied and expanded to embrace the use of real-valued orthogonal wavelet basis, which we show to yield satisfactory results only under the condition of adding a safety zone in wavelet space. We also apply the complex-valued wavelet based method to the 2D Euler equation problem, showing that it is able to filter the resonances in this case as well.