**
Discrete Approximation of the Riemann Problem
for the Viscous Burgers Equation
**

In this paper we consider discrete approximations of a Dirichlet
problem for the quasilinear parabolic equation

that is, the viscous Burgers equation.
The singular perturbation parameter e
takes arbitrary values from the half-interval (0,1].
The initial condition has a discontinuity of the first kind at
the point *S*^{*}=(0,0)
such that j_{0}(+0)-j_{0}(-0)>0,
where *u*(*x*,0)=j_{0}(*x*);
thus, we have the Riemann problem. For such a problem we construct
special finite difference schemes controlled by the parameter
e
and by the type of the singularities, which the solution
*u*(*x*,*t*)
exhibits. The discrete solution for this problem is shown to converge
uniformly with respect to the parameter
e
in a uniform grid metric.