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In this paper, efficient alternating direction implicit (ADI) schemes areproposed to solve three-dimensional heat equations with irregular boundariesand interfaces. Starting from the well-known Douglas-Gunn ADI scheme, amodified ADI scheme is constructed to mitigate the issue of accuracy loss insolving problems with time-dependent boundary conditions. The unconditionalstability of the new ADI scheme is also rigorously proven with the Fourieranalysis. Then, by combining the ADI schemes with a 1D kernel-free boundaryintegral (KFBI) method, KFBI-ADI schemes are developed to solve the heatequation with irregular boundaries. In 1D sub-problems of the KFBI-ADI schemes,the KFBI discretization takes advantage of the Cartesian grid and preserves thestructure of the coefficient matrix so that the fast Thomas algorithm can beapplied to solve the linear system efficiently. Second-order accuracy andunconditional stability of the KFBI-ADI schemes are verified through severalnumerical tests for both the heat equation and a reaction-diffusion equation.For the Stefan problem, which is a free boundary problem of the heat equation,a level set method is incorporated into the ADI method to capture thetime-dependent interface. Numerical examples for simulating 3D dendriticsolidification phenomenons are also presented.

ADI schemes for heat equations with irregular boundaries and interfaces in 3D with applications