The Embedded Boundary Integral Method (EBI) for the Incompressible Navier-Stokes equations

We present a new method for the solution of the unsteady incompressible Navier-Stokes equations. Our goal is to achieve a robust and scalable methodology for two and three dimensional incompressible flows. The discretization of the Navier-Stokes operator is done using boundary integrals and structured-grid finite elements. We use finite-differences to advance the equations in time. The convective term is discretized via a semi-Lagrangian formulation which not only results in a spatial constant-coefficient (modified) Stokes operator, but in addition is unconditionally stable. The Stokes operator is inverted by a double-layer boundary integral formulation. Domain integrals are computed via finite elements with appropriate forcing singularities to account for the irregular geometry. We use a velocity-pressure formulation which we discretize with bilinear elements (Q1-Q1), which give equal order interpolation for the velocities and pressures. Stabilization is used to circumvent the div-stability condition for the pressure space. The integral equations are discretized by Nyström’s method. For the specific approximation choices the method is second order accurate. Our code is built on top of PETSc, an MPI based parallel linear algebra library. We will present numerical results and discuss the performance and scalability of the method in two dimensions.