We investigate the convergence of difference schemes for the one-dimensional heat equation when the coefficient at the time derivative (heat capacity) is c (x) = 1 + Kδ (x − ξ). K = const > 0 represents the magnitude of the heat capacity concentrated at the point x = ξ. An abstract operator method is developed for analyzing this equation. Estimates for the rate of convergence in special discrete energetic Sobolev's norms, compatible with the smoothness of the solution are obtained. Subject Classification (1991): 65M06, 65M12, 65M15 Mathematics

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A diffusion equation with nonlinear localized chemical reactions is considered in this paper. As a result of the reactions, although the equation is parabolic, the derivatives of the solution are discontinuous across the interfaces (local sites of reactions). A second-order accurate immersed interface method is constructed for the diffusion equation involving interfaces. The new method is more accurate than the standard approach and it does not require the interfaces to be grid points. Several experiments that confirm second-order accuracy are presented. The efficiency of the proposed algorithm is also demonstrated for solving blow up problems. The proposed technique could be extended for construction of efficient numerical algorithms on uniform grids for the present equations with moving interfaces [9] but more analysis is required.

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