A Fourth-Order Cartesian Grid Method with Fft Acceleration for Elliptic and Parabolic Problems on Irregular Domains and Arbitrarily Curved Boundaries

Abstract

This work aims to develop a fourth-order Cartesian grid finite difference method in two dimensions for solving elliptic and parabolic problems over an irregular domain with an arbitrarily curved boundary, under the assumptions that the boundary is C1 continuous and the grid resolution is fine enough inside the domain. The proposed Augmented Matched Interface and Boundary (AMIB) will inherit its predecessor’s speed and accuracy advantages, such as maintaining the fast Fourier-transform (FFT) efficiency and being fourth-order of accurate in handling any boundary conditions (Dirichlet, Neumann, Robin, or their combinations). To accommodate an arbitrarily curved boundary, the proposed next-generation AMIB method features two significant improvements. First, an adaptive ray-casting Matched Interface and Boundary (MIB) scheme is developed. It solves much more complex geometry by producing fictitious values adaptively through an iterative process in which a new fictitious value may depend on previously calculated fictitious values. Second, several stabilizers have been designed to ensure the stability of the AMIB method, which include a new preconditioner for the augmented linear system and proper grid selection requirements in interpolation and derivative approximations of the MIB scheme. Numerical experiments have been conducted to validate the proposed AMIB method for solving boundary and initial value problems with sharply curved and/or moving boundaries.

Keywords: Elliptic boundary value problem, Parabolic initial-and-boundary value problems, Irregular domains, High order finite difference, Augmented Matched interface and boundary (AMIB), Fast Fourier Transform (FFT)