Matched interface and boundary (MIB) for the implementation of boundary conditions in high‐order central finite differences

Abstract

SUMMARYHigh-order central finite difference schemes encounter great difficulties in implementing complex boundary conditions. This paper introduces the matched interface and boundary (MIB) method as a novel boundary scheme to treat various general boundary conditions in arbitrarily high-order central finite difference schemes. To attain arbitrarily high order, the MIB method accurately extends the solution beyond the boundary by repeatedly enforcing only the original set of boundary conditions. The proposed approach is extensively validated via boundary value problems, initial-boundary value problems, eigenvalue problems, and high-order differential equations. Successful implementations are given to not only Dirichlet, Neumann, and Robin boundary conditions, but also more general ones, such as multiple boundary conditions in high-order differential equations and time-dependent boundary conditions in evolution equations. Detailed stability analysis of the MIB method is carried out. The MIB method is shown to be able to deliver high-order accuracy, while maintaining the same or similar stability conditions of the standard high-order central difference approximations. The application of the proposed MIB method to the boundary treatment of other non-standard high-order methods is also considered.

Analysis of the diffuse-domain method for solving PDEs in complex geometries

“…However, most of these methods typically require tools not frequently available in standard finite element and finite difference software packages. Examples of such approaches include the extended and composite finite element methods (e.g., [31,12,23,13,32,55,7,4]), immersed interface methods (e.g., [40,43,60,44,65]), virtual node methods with embedded boundary conditions (e.g., [3,73,34]), matched interface and boundary methods (e.g., [71,68,69,67,72]), modified finite volume/embedded boundary/cut-cell methods/ghost-fluid methods (e.g., [27,36,19,25,26,35,47,70,48,37,46,64,49,9,10,52,53,33,63]). In another approach, known as the fictitious domain method (e.g., [28,29,56,45]), the original system is either augmented with equations for Lagrange multipliers to enforce the boundary conditions, or the penalty method is used to enforce the boundary condi-tions weakly.…”

High order matched interface and boundary methods for the Helmholtz equation in media with arbitrarily curved interfaces

“…An iterative procedure is commonly employed in the MIB schemes [40,42,41,38,39] Denote the finite difference (FD) weight vector of these two stencils differentiating at C 0 to be, respectively, W À k and W þ k . Here the subscript k represents interpolation (k ¼ 0) and the first order derivative approximation (k ¼ 1).…”

“…In the proposed MIB method and the previous MIB schemes [40,42,38,39,37,41], the use of fictitious nodes is an important step in enforcing the jump/boundary conditions. In terms of using fictitious points or ghost cells, the MIB shares some similarities with the ghost fluid method (GFM), originally developed by Fedkiw et al [9] for treating contact discontinuities in the inviscid Euler equations.…”

“…The scalar MIB method has been generalized to treat curved interfaces for solving the Poisson equation [42]. Recently, the application of the MIB method as a fictitious domain boundary closure scheme for high order finite differences has been considered in [37,41].…”

Free vibration analysis of multiple-stepped beams by the differential quadrature element method

“…With the matched interface and boundary (MIB) method [21], a novel boundary scheme to treat various general boundary conditions, the DSC should have no difficulty to handle the multiple boundary conditions of beam structures. However, the DSC suffers similar difficulties encountered by the DQM in analyzing the free vibration problem of multiple-stepped beams.…”