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Monotone schemes for degenerate and non-degenerate parabolic problems Limic, Mirna

Abstract

This thesis analyses several monotone schemes for degenerate second order parabolic PDEs over two-dimensional domains which is in most cases equal (0, l )2 . For such problems the degeneracy means that in addition to the axis spanned by the standard basis vectors, there exists another pair of orthogonal cordinate axes such that the spatial difference operator is of the second order in one of the directions, and of the first order in the remaining directions. The direction of one axis along which the spatial difference operator is of the second order we call the direction of diffusion. The thesis considers only constant coefficient PDE's and therefore the degeneracy can be easily determined by solving the eigenvalue problem for diffusion tensor. We analyse the impact of the direction of diffusion on the convergence of a scheme. Previous work on a second order elliptic problem has shown that central differences taken in the direction of diffusion produce a convergent scheme, wherease disalignment between the two result in non-convergent schemes. One of our aims was to check the validity of this finding and possibly improve the construction of schemes. As a result we present a novel approach to building monotone schemes from the diffusion tensor by taking spatial step sizes of different length in order to align the second order central differences with the direction of diffusion. We give a step by step algorithm and supply all of the findings. Our findings are based on MATLAB m file, and C language implementations.

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