The convergence of the GRP scheme

Pages: 1 - 27,
Volume 23,
Issue 1/2,
January/February
2009 doi:10.3934/dcds.2009.23.1

Matania Ben–Artzi - Institute of Mathematics, the Hebrew University of Jerusalem, 91904, Israel (email)

Joseph Falcovitz - Institute of Mathematics, the Hebrew University of Jerusalem, 91904, Israel (email)

Jiequan Li - School of Mathematical Sciences, Capital Normal University, 100037, Beijing, China (email)

Abstract:
This paper deals with the convergence of the second-order GRP
(Generalized Riemann Problem) numerical scheme to the entropy
solution for scalar conservation laws with strictly convex fluxes.
The approximate profiles at each time step are linear in each cell,
with possible jump discontinuities (of functional values and slopes)
across cell boundaries. The basic observation is that the discrete
values produced by the scheme are **exact averages
** of an **approximate conservation law**
, which enables the use of properties
of such solutions in the proof. In particular, the
“total-variation" of the scheme can be controlled, using analytic
properties. In practice, the GRP code allows “sawteeth" profiles
(i.e., the piecewise linear approximation is not monotone even if
the sequences of averages is such). The “reconstruction" procedure
considered here also allows the formation of “sawteeth" profiles,
with an hypothesis of “Godunov Compatibility", which limits the
slopes in cases of non-monotone profiles. The scheme is proved to
converge to a weak solution of the conservation law. In the case of
a monotone initial profile it is shown (under a further hypothesis
on the slopes) that the limit solution is indeed the entropy
solution. The constructed solution satisfies the “finite
propagation speed", so that no rarefaction shocks can appear in
intervals such that the initial function is monotone in their domain
of dependence. However, the characterization of the limit solution
as the unique entropy solution, for general initial data, is still
an open problem.

Keywords: Hyperbolic conservation laws, the GRP scheme, Convergence, TVD, entropy

Mathematics Subject Classification: Primary: 65M06, 65M12; Secondary: 35L65, 35L67

Received: December 2007;
Revised:
May 2008;
Available Online: September 2008.