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

TL;DR

This paper introduces efficient ADI schemes for 3D heat equations with irregular boundaries, combining stability, accuracy, and applications to complex free boundary problems like dendritic solidification.

Contribution

It develops modified ADI schemes with proven stability, integrating boundary integral methods for irregular boundaries, and applies them to complex free boundary problems.

Findings

The new ADI scheme is unconditionally stable and second-order accurate.

KFBI-ADI schemes efficiently handle irregular boundaries on Cartesian grids.

Numerical tests demonstrate effectiveness in simulating dendritic solidification.

Abstract

In this paper, efficient alternating direction implicit (ADI) schemes are proposed to solve three-dimensional heat equations with irregular boundaries and interfaces. Starting from the well-known Douglas-Gunn ADI scheme, a modified ADI scheme is constructed to mitigate the issue of accuracy loss in solving problems with time-dependent boundary conditions. The unconditional stability of the new ADI scheme is also rigorously proven with the Fourier analysis. Then, by combining the ADI schemes with a 1D kernel-free boundary integral (KFBI) method, KFBI-ADI schemes are developed to solve the heat equation with irregular boundaries. In 1D sub-problems of the KFBI-ADI schemes, the KFBI discretization takes advantage of the Cartesian grid and preserves the structure of the coefficient matrix so that the fast Thomas algorithm can be applied to solve the linear system efficiently. Second-order accuracy and unconditional stability of the KFBI-ADI schemes are verified through several numerical 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 the time-dependent interface. Numerical examples for simulating 3D dendritic solidification phenomenons are also presented.