Hemker, P. W.; Shishkin, G. I. Approximation of parabolic PDEs with a discontinuous initial condition. (English) Zbl 0835.65101 East-West J. Numer. Math. 1, No. 4, 287-302 (1993). Summary: We consider a Dirichlet problem for a parabolic partial differential equation with a discontinuous initial condition. The boundary condition at \(t= 0\) is assumed to have a discontinuity of the first kind. Due to the singularity of the solution in the neighbourhood of the discontinuity, the usual discretization methods do not yield convergence in the \(\ell^\infty\)-norm in the entire domain of definition. Therefore, in order to handle the singularity an adapted scheme is constructed. We use a specially fitted difference operator on a regular rectangular grid. Such a difference scheme converges in the discrete \(\ell^\infty\)-norm on the whole uniform grid. For a model problem, numerical experiments with the classical and the specially fitted schemes are compared and discussed. Cited in 7 Documents MSC: 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35K20 Initial-boundary value problems for second-order parabolic equations Keywords:finite difference methods; uniform convergence; parabolic boundary value problems; Dirichlet problem; discontinuous initial condition; convergence; numerical experiments PDFBibTeX XMLCite \textit{P. W. Hemker} and \textit{G. I. Shishkin}, East-West J. Numer. Math. 1, No. 4, 287--302 (1993; Zbl 0835.65101)