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We discuss the impact of modal filtering in Legendre spectral methods, both on accuracy and stability. For the former, we derive sufficient conditions on the filter to recover high order accuracy away from points of dis- continuity. Computational results confirm that less strict necessary conditions appear to be adequate. We proceed to discuss a instability mechanism in poly- nomial spectral methods and prove that filtering suffices to ensure stability. The results are illustrated by computational experiments. While the advantages of the use of spectral and pseudospectral methods for solv- ing partial differential equations with smooth solutions are widely acknowledged (3, 18, 8), the prospects of using such methods for problems with nonsmooth solu- tions remain more controversial. This can be attributed mainly to the appearance of the Gibbs phenomenon, appearing when approximating nonsmooth solutions us- ing polynomials, and the often detrimental effect this has on the stability of the computational scheme. Thus, it is often perceived that spectral methods are too sensitive and lack robustness to allow the modeling of problems of realistic com- plexity which, by their very nature, most often are dominated by under-resolved and unresolved dynamics. The literature is rich with ideas for overcoming this lack of robustness, all cen- tered around the idea of introducing sufficient dissipation of the high modes without sacrificing the accuracy. The exact process of doing so is less obvious as too much dissipation clearly destroys the accuracy of the solution. Recently, the use of spectral filters to regain robustness has received considerable attention due to the effectiveness of this approach and the low computational cost of applying such filters compared to alternatives, e.g., limiting. The filters are characterized by modifying the expansion coefficients of the solution, thus leading to a global modification of the solution. For Fourier spectral and pseudospectral methods, early results showed the promise of this for linear problems (21) and with experimental evidence for nonlin- ear problems (20). The first rigorous analysis of the approximation properties of the filtered expansion is offered in (27), showing the potential of recovering spectral

Filtering in Legendre spectral methods