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 j0(+0)-j0(-0)>0,
thus, we have the Riemann problem. For such a problem we construct
special finite difference schemes controlled by the parameter
and by the type of the singularities, which the solution
exhibits. The discrete solution for this problem is shown to converge
uniformly with respect to the parameter
in a uniform grid metric.