A Second-Order Boundary Condition Capturing Method for Solving the Elliptic Interface Problems on Irregular Domains

Abstract

A second-order boundary condition capturing method is presented for the elliptic interface problem with jump conditions in the solution and its normal derivative. The proposed method is an extension of the work in Liu et al. (J Comput Phys 160(1):151-178, 2000) to a higher order. The motivation of proposed method is that the approximated value at the interface can be reconstructed by proper interpolation based on the level set representation from Gibou et al. (J Comput Phys 176(1):205-227, 2002). A second-order accurate method is constructed, both in the solution and its gradient, using second-order finite difference approximation. Several numerical results demonstrate that the proposed method is indeed second-order accurate in the solution and its gradient in the L 2 and L ∞ norms.

A second-order accurate semi-Lagrangian method for convection-diffusion equations with interfacial jumps

“…2.2 A second-order ghost fluid method by Cho et al [3] In this subsection, we briefly review the second-order ghost fluid method proposed by Cho et al [3], focusing on the core idea of approximating u at the interface. Consider a jump condition…”

“…Figure 4.1 shows the corresponding solutions on the 160 × 160 grid. Though the SL-BDF2 also discretizes diffusive terms using a ghost-fluid method [3], overall accuracy is first-order. Besides, we can see that the SL-GFM yields second-order convergence.…”

“…While the GFM is simpler to implement than IIM, it loses accuracy due to ignorance of tangential jump conditions. To address this issue, first-and second-order extensions of GFM were developed in [6] and [3,8] for solving elliptic interface problems. Moreover, there are other various methods for the interface problems on Cartesian grids.…”

“…In this paper, we introduce a second-order accurate semi-Lagrangian method for moving interface problems (1.1)-(1.2) on a Cartesian grid. The proposed method follows the framework of the secondorder ghost fluid method [3] to capture jump conditions and the level-set method [23] to track the moving interface. The main idea of this study is to incorporate jump conditions into the discretization of convection terms and diffusion terms.…”

Fully implicit and accurate treatment of jump conditions for two-phase incompressible Navier-Stokes equation

“…A jump condition equivalent to (2) is first derived through the replacement of normal derivatives with a linear combination of Cartesian and tangential derivatives, as was done for the elliptic interface problem in [5,7]. The normal and tangent vectors of the interface are then defined as n = (n x , n y ) and τ = (−n y , n x ), respectively.…”