A Velocity Decomposition Approach for Solving the Immersed Interface Problem with Dirichlet Boundary Conditions
In a previous study, we presented a second-order accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness (Beale and Layton, J Comput Phys 228:3358–3367, 2009). The fluid flow was described by the Navier-Stokes equations with periodic boundary conditions, and the deformation of the moving interface exerts a singular force onto the fluid. In this study, we extend that method to Dirichlet boundary conditions. We decompose the velocity into three parts: a “Stokes” part, a “regular” part, and a “boundary correction” part. The “Stokes” part is determined by the Stokes equations and the singular interfacial force. The Stokes solution is obtained using the immersed interface method, which gives second-order accurate values by incorporating known jumps for the solution and its derivatives into a finite difference method. The regular part of the velocity is given by the Navier-Stokes equations with a body force resulting from the Stokes part, and with periodic boundary conditions. The regular velocity is obtained using a time-stepping method that combines the semi-Lagrangian method with the backward difference formula. Because the body force is continuous, jump conditions are not necessary. The boundary correction solution is described by the unforced Navier-Stokes equations, with Dirichet boundary conditions given by the difference between the Dirichlet boundary conditions of the overall Navier-Stokes solution, and the boundary values of the Stokes and regular velocities. Because the boundary correction solution is sufficiently smooth, jump conditions are also not necessary. Numerical results exhibit approximately second-order accuracy in time and space.