## Viscosity Approximations to Multidimensional
Scalar Conservation Laws

#### Authors: Gui-Qiang Chen, Qiang Du, and Eitan Tadmor

#### Title: Viscosity Approximations to Multidimensional Scalar
Conservation Laws

**Abstract**

We study the spectral viscosity (SV) method in the context of
multidimensional scalar conservation laws with periodic boundary
conditions. We show that the spectral viscosity, which is sufficiently
small to retain the formal spectral accuracy of the underlying Fourier
approximation, is large enough to enforce the correct amount of entropy
dissipation (which is otherwise missing in the standard Fourier method).
Moreover, we prove that because of the presence of the spectral viscosity,
the truncation error in this case becomes spectrally small,
{\it independent} of whether the underlying solution is smooth or not.
Consequently, the SV approximation remains uniformly bounded and converges
to a measure-valued solution satisfying the entropy condition,
that is, the unique entropy solution. We also show that the SV solution
has a bounded total variation, provided that the total variation of the
initial data is bounded, thus confirming its strong convergence to
the entropy solution. We obtain an $L^{1}$ convergence rate of the
usual optimal order one-half.

This article has appeared in:

*Mathematics of Computation*, **vol. 6**, pages 629-643 (1993)

This paper is available in the following formats:

A closely related paper is Change me.

**Author Address**
Gui-Qiang Chen
Department of Mathematics
Northwestern University
Evanston, IL 60208-2730
gqchen@math.nwu.edu