# Preprint 2007-016

# A convergent finite difference method for a nonlinear variational wave equation

## H. Holden, K. H. Karlsen and N. H. Risebro

**Abstract:**
We establish rigorously convergence
of a semi-discrete upwind scheme
for the nonlinear variational wave equation
*u*_{tt}−*c*(*u*)(*c*(*u*)*u*_{x})_{x}=0
with
*u*|_{t=0}=*u*_{0}
and
*u*_{t}|_{t=0}=*v*_{0}.
Introducing Riemann invariants
*R*=*u*_{t}+*cu*_{x}
and
*S*=*u*_{t}−*cu*_{x},
the variational wave equation is equivalent to
*R*_{t}−*cR*_{x}=*~c*(*R*^{2}−*S*^{2})
and
*S*_{t}+*cS*_{x}=−*~c*(*R*^{2}−*S*^{2})
with *~c*=*c*′/(4*c*).
An upwind scheme is defined for this system.
We assume that the the speed *c* is positive,
increasing and both *c* and its derivative
are bounded away from zero and that
*R*|_{t=0}, *S*|_{t=0}∈*L*^{1}(**R**)∩*L*^{3}(**R**)
are nonpositive.
The numerical scheme is illustrated on several examples.