# Preprint 2009-046

# Error bound between monotone difference schemes and their modified equations

## Zhen-huan Teng

**Abstract:**
It is widely believed that
if monotone difference schemes are applied
to the linear convection equation with discontinuous initial data,
then solutions of the monotone schemes are closer
to solutions of their parabolic modified equations
than that of the original convection equation.
We will confirm the conjecture in this paper.
It is well known that solutions of the monotone schemes
and their parabolic modified equations
approach discontinuous solutions
of the linear convection equation
at a rate only half in the
L^{1}-norm.
We will prove that the error bound
between solutions of the monotone schemes
and that of their modified equations is *order one*
in the L^{1}-norm.
Therefore the conclusion shows that
the monotone schemes solve the modified equations
more accurately than the original convection equation
even if the initial data is discontinuous.
As a consequence of the main result,
we will show that the half-order rate of convergence
for the monotone schemes to the convection equation
is the best possible.