# Preprint 2007-011

# An Engquist–Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections

## Raimund Bürger, Kenneth H. Karlsen and John D. Towers

**Abstract:**
We consider scalar conservation laws with the spatially varying flux
*H*(*x*)*f*(*u*)+(1−*H*(*x*))*g*(*u*),
where *H*(*x*) is the Heaviside function
and *f*
and *g*
are smooth nonlinear functions.
Adimurthi, Mishra, and Veerappa Gowda
[*J. Hyperbolic Differ. Equ.* **2**:783–837, 2005]
pointed out that such a conservation law
admits many *L*^{1}
contraction semigroups,
one for each so-called connection
(*A*,*B*).
Here we define entropy solutions of type
(*A*,*B*)
involving Kružkov-type entropy
inequalities that can be adapted to any fixed connection
(*A*,*B*).
It is proved that these entropy inequalities imply the
*L*^{1}
contraction property for
*L*^{∞} solutions,
in contrast to the “piecewise smooth”
setting of Adimurthi et al.
For a fixed connection, these entropy inequalities
include a single adapted entropy of the type
used by Audusse and Perthame
[*Proc. Roy. Soc. Edinburgh A* **135**:253–265, 2005].
We prove convergence of a new difference scheme that approximates
entropy solutions of type (*A*,*B*)
for any connection (*A*,*B*)
if a few parameters are varied.
The scheme relies on a modification of the standard
Engquist–Osher flux,
is simple as no 2×2 Riemann solver is involved,
and is designed such that the steady-state solution connecting
*A* to *B* is preserved.
In contrast to most analyses of similar problems,
our convergence proof is not based on the singular mapping
or compensated compactness methods,
but on standard *BV* estimates
away from the flux discontinuity.
Some numerical examples are presented.