Efficient algorithms for solving three dimensional parabolic interface problem with variable coefficients

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The dissertation consists of two parts, in the first part, a new matched alternating direction implicit (ADI) method is proposed for solving three-dimensional (3D) parabolic interface problems with discontinuous jumps, piecewise constant diffusion coefficients and complex interfaces. This scheme inherits the merits of its ancestor of two-dimensional problems, while possesses several novel features, such as a non-orthogonal local coordinate system for decoupling the jump conditions, two-side estimation of tangential derivatives at an interface point, and a new Douglas-Rachford ADI formulation that minimizes the number of perturbation terms, to attack more challenging 3D problems. In time discretization, this new ADI method is found to be first order and stable in numerical experiments. In space discretization, the matched ADI method achieves a second order of accuracy based on simple Cartesian grids for various irregularly-shaped surfaces and spatial-temporal dependent jumps. Computationally, the matched ADI method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3D parabolic equations, in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom N, i.e., O(N). Furthermore, unlike iterative methods, the ADI method is an exact or non-iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed N. Therefore, the proposed matched ADI method provides an efficient tool for solving 3D parabolic interface problems. In the second part, instead of constant diffusion coefficients, improved schemes for variable diffusion coefficient are also performed in the work. A comparison of proposed ADI method with different other time splitting methods, including locally one-dimensional implicit Euler(LOD-IE), locally one-dimensional Crank-Nicolson(LOD-CN) and Trapezoidal Splitting(TS) method will be implemented, coupled with different variation of matched interface and boundary (MIB) method in spatial discretization. These large scale computational studies facilitate the further development of matched ADI algorithms for 3D parabolic interface problems.

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