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

Abstract

It is well known that solutions to the FourierGalerkin 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.

Stability and spectral convergence of Fourier method for nonlinear problems: on the shortcomings of the $$2/3$$ 2 / 3 de-aliasing method

“…Each one of these scenarios implies that u m contains spurious oscillations which are noticeable throughout the computational domain, in agreement with the numerical evidence observed in [Tad89]. We note that this type of nonlinear instability applies to both, the 2/3 method and in particular, the spectral Fourier method and we refer in this context to the recent detailed study in [RFNM11,PNFS13] and the refernces therein.…”

Suppressing thermalization and constructing weak solutions in truncated inviscid equations of hydrodynamics: Lessons from the Burgers equation

“…However the precise mechanism by which solutions thermalize was discovered later by Ray et al [17], who showed that thermalization was triggered through a resonant-wave-like interaction leading to localized structures christened tygers (see also Refs. [18][19][20]).…”

“…However the process and mechanisms of thermalization was best understood by resorting to the 1D Burgers equation [3,14,17,20]; in the same spirit, we now outline and present results for the efficacy of the tyger purging method. At the end of this paper, we will return briefly to its applicability to the problem of the 3D Euler equation as well as contrast our approach with wavelet-based filtering techniques [18].…”

“…We refer the reader to Refs. [3,14,[17][18][19][20] for more details and the theory of this process of thermalization.…”

“…This however has the drawback that we would end up solving not the inviscid equation but its viscous form and for higher orders of the hyperviscosity-which is similar in spirit to the idea of purging-the solutions thermalize [10,12,32]. Another approach is due to Pereira et al [18] who showed that a waveletbased filtering technique also leads to a suppression of the resonances leading to tygers. However, such an approach has the limitation, as mentioned by the authors themselves, that the dual operations of filtering and truncation at every time step do not commute.…”