Lefschetz Center for Dynamical Systems Seminar
No Seminar, Columbus Day Holiday
Brown Analysis Seminar
Scientific Computing Seminar
Abstract: The problem of the Gibbs phenomenon arises when high-order methods are used to simulate flows with discontinuities, such as shocks and interfaces. Two methodologies are described for handling this problem: (1) postprocessing the flow fields with a Gegenbauer or Jacobi polynomial reconstruction (originally proposed by Gottlieb & Shu), and (2) using artificial dissipation terms to damp Gibbs oscillations near discontinuous features.
The first methodology is difficult to implement in practice and can still lead to unphysical flow behavior when oscillations are too large. The second methodology is based on high-order compact and spectral schemes, and it can be used to compute multicomponent turbulent flows at any Mach number. In this case, filters are employed to stabilize the numerical integration and high-order artificial transport coefficients are introduced to control the Gibbs oscillations. The equations and numerical scheme are formulated such that, under grid refinement, the method approaches DNS.
The method is evaluated for flows in 1, 2, and 3 dimensions, including comparisons with Godunov, ALE, and CENO schemes. The compact/spectral method is found to offer significant benefits over the low-order, highly diffusive schemes like CENO, but comparisons with the Godunov and ALE methods yield mixed conclusions. The dissipative character of the filter and artificial terms appears to be of little consequence for strongly forced flows which evolve over short periods of time; however, the dissipation is more noticeable for unforced flows which evolve over long periods of time.
PDE Seminar
Department of Mathematics Colloquium
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