| TI: |
Title
Enhanced spectral viscosity approximations for conservation laws
|
| AU: |
Author
Gelb, A; Tadmor, E |
| AF: |
Author Affiliation
Arizona State University, USA; UCLA, USA |
| SO: |
Source
Applied Numerical Mathematics, vol. 33, no. 1-4, pp. 3(18), May
2000 |
| IS: |
ISSN
0168-9274 |
| AB: |
Abstract
In this paper 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 (Tadmor, 1989) 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-O(1/ N ) 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 a priori 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 (Gelb and
Tadmor, 1999). 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. |
| PY: |
Publication Year
2000 |
| CP: |
Country of Publication
Netherlands |
| DE: |
Descriptors
spectral methods; conservation laws; approximation |
| CL:: |
Classification
Mathematics of computing; Numerical analysis |
| SH: |
Shelfmark
British Library: 1576.234000 |
| AN: |
Accession Number
44_6948 |
