Simple Second-Order Finite Differences for Elliptic PDEs with Discontinuous Coefficients and Interfaces
For researchers solving multi-phase flow or fluid-structure interaction problems, this method simplifies the numerical treatment of irregular interfaces while maintaining high accuracy.
The paper presents a new finite difference method for elliptic PDEs with discontinuous coefficients and interfaces, achieving second-order accuracy with a simple Cartesian grid formulation. Numerical tests on the Poisson equation demonstrate second-order convergence in 2D and 3D.
In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise due to changes in material properties at an immersed interface or embedded boundary, which may have an irregular shape. Consequently, the solution and its gradient can be discontinuous, and numerical methods can be difficult to design. Here a new method is presented and analyzed, using a simple formulation of one-dimensional finite differences on a Cartesian grid, allowing for a relatively easy setup for one-, two-, or three-dimensional problems. The method preserves a sharp interface with discontinuous solutions, obtained from a small number of iterations (approximately five) of solving a symmetric linear system with updates to the right- hand side. Second-order accuracy is rigorously proven in one spatial dimension and demonstrated through numerical examples in two and three spatial dimensions. The method is tested here on the variable-coefficient Poisson equation, and it could be extended for use on time-dependent problems of heat transfer, fluid dynamics, or other applications.