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The Jump Splice for Elliptic Interface Problems and the Incompressible Navier-Stokes Equations

  • Author(s): Preskill, Benjamin David
  • Advisor(s): Sethian, James A
  • et al.
Abstract

We present a general framework for accurately evaluating finite difference operators in the presence of known discontinuities across an interface. Using these techniques, we develop simple-to-implement, second-order accurate methods for elliptic problems with interfacial discontinuities and for the incompressible Navier-Stokes equations with singular forces. To do so, we first establish an expression relating the derivatives being evaluated, the finite difference stencil, and a compact extrapolation of the jump conditions. By representing the interface with a level set function, we show that this extrapolation can be constructed using dimension- and coordinate-independent normal Taylor expansions with arbitrary order of accuracy. Our method is robust to non-smooth geometry, permits the use of symmetric positive-definite solvers for elliptic equations, and also works in 3D with only a change in finite difference stencil. We rigorously establish the convergence properties of the method and present extensive numerical results. In particular, we show that our method is second-order accurate for the incompressible Navier-Stokes equations with surface tension.

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