Enhanced Spectral Viscosity Approximations for Conservation Laws
Abstract
In this talk we construct, analyze and implement a new procedure
for the spectral approximations of nonlinear conservation laws.
It is well known that using spectral methods for nonlinear
conservation laws will result in the formation of the Gibbs phenomenon
once spontaneous shock discontinuities appear in the solution.
These spurious oscillations will in turn
lead to loss of resolution and render the standard
spectral approximations unstable.
The Spectral Viscosity (SV-) method was developed to
stabilize the spectral method by adding a spectrally small
amount of high-frequencies diffusion carried out in the dual space.
The resulting SV-approximation is stable without sacrificing
spectral accuracy.
The SV-method recovers a spectrally accurate approximation
to the projection of the entropy solution; the
exact projection, however, is at best a first order approximation to
the exact solution as a result of the
formation of the shock discontinuities.
The issue of spectral resolution is addressed by
post-processing the SV-solution to remove
the spurious oscillations at the discontinuities, as well
as increase the first-order accuracy away from the
shock discontinuities.
Successful post-processing methods have been developed to eliminate
the Gibbs phenomenon and recover spectral accuracy
for the SV-approximation. However,
such reconstruction methods require apriori knowledge
of the locations of the shock discontinuities.
Therefore, the detection of these discontinuities is essential to
obtain an overall spectrally accurate solution.
To this end, we employ the recently constructed
enhanced edge detectors based on appropriate
concentration factors.
Once the edges of these discontinuities are identified,
we can utilize a post-processing reconstruction method,
and show that the post-processed SV-solution
recovers the exact entropy solution with remarkably high-resolution.
We apply our new numerical method, the Enhanced SV-method,
to two numerical examples, the scalar periodic
Burgers' equation and the one-dimensional system of Euler
equations of gas dynamics. Both approximations exhibit
high accuracy and resolution to the exact entropy solution.
