The numerical solution of a singularly perturbed problem for quasilinear parabolic differential equation (Q1203851)

The Dirichlet initial-boundary value problem for a quasilinear parabolic differential equation of second order, in one space variable, where the coefficient of the second space derivative contains a multiplicative small parameter \(\varepsilon\), is considered. The authors describe and investigate in detail a three-level difference scheme with appropriately chosen nonuniform grid in the spatial variable which is three-diagonally linear in the highest time level. They show this scheme to be uniformly (with respect to \(\varepsilon)\) convergent in the discrete \(L^ 2\)-norm (in a compact time interval). In fact, they obtain convergence of order \(\tau^ 2+h^{2/3}\) with \(h\) as maximal space step and \(\tau\) as time step of the discretization (under suitable smoothness assumptions). They illustrate their results by displaing two nice numerical examples.