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...nuous coefficients and singular forces. It has been successfully applied to moving boundary problems, for example for the Stokes problem with elastic interfaces [32] and for the Navier-Stokes problem =-=[35]-=-. If the singular forces are known then the jumps are known and TESCs can be computed explicitly. For discontinuous coefficients IIM modifies the stencils for points close to the boundary in order to ...

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...i [23] for solving elliptic equations, and later extended to Stokes flow with elastic boundaries or surface tension [24]. The method was developed further for the Navier-Stokes equations in Li et al. =-=[25]-=-, Lee [22] and Le et al. [19] for problems with flexible boundaries. The method was also used by Calhoun [6] and Li et al. [26] for solving the two-dimensional streamfunction-vorticity equations on ir...

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...apply our finite-difference scheme to any of these methods. For one example of how the immersed interface method can be applied to a primitive variable formulation of the Navier–Stokes equations, see =-=[29]-=-. APPENDIX A.1. Derivation of Coefficients for the Immersed Interface Method Here we provide formulas for the coefficients needed to construct the matrix S in (33) and for extending the solution ω to ...

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... the advection or the Maxwell equations [4] with interfaces. 29 The extension of the ESIM to nonlinear problems is a challenging project. We think to CFD problems, such as the Navier-Stokes equations =-=[15]-=- or the Euler equations with moving interfaces. We have especially in mind the numerical treatment of material interfaces in multicomponent flows. The ”Ghost Fluid Method” of Fedkiw and al. [5] is eff...

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...s described briefly in section 3.1. The extension of this approach to flows with non-zero Reynolds numbers (but matched viscosities and densities) is straightforward using the techniques described in =-=[27, 37, 30]-=- for the IIM. The extension to flows with variable viscosity has also been recently developed [36]. Finally, the nondimensional form of the surfactant equation is identical to Eq. (9) with Ds replaced...

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...,1,8,2]. Among these developments, Wiegmann and Bube [55,56] developed an explicit-jump immersed interface method for the special cases where the explicit jump conditions of variables and derivatives ([u], [ux], [uxx], etc.) are known. Though in a simpler explicit form, the explicit-jump immersed interface method is not applicable to the general jump conditions given by (2). The immersed interface methods have been applied to the Stokes flow with elastic boundaries or surface tension [25], Hele–Shaw flow [17], incompressible flow based on the Navier–Stokes equations with singular source terms [29,5,23], and nonlinear problems in magneto-rheological fluids [19]. Despite these applications, the immersed interface methods are often difficult to apply to complex two or three-dimensional two-phase flow problems. In order to maintain a second-order accuracy, it is necessary to obtain jump conditions at the interface for flow variables and their first and second derivatives. For the Navier–Stokes equations with an interface of discontinuity, it is easy to derive the physical jump conditions for flow variables and their first derivatives across the interface. But it is difficult to obtain jump cond...