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Approximation of parabolic PDEs with a discontinuous initial condition. (English) Zbl 0835.65101

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.

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
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