Christoph Borgers 
Lagrangian Fractional Step Method for the Incompressible Navier-Stokes Equations [PDF ebook] 
October 1985

Whilst the greatest effort has been made to ensure the quality of this text, due to the historical nature of this content, in some rare cases there may be minor issues with legibility. We develop a modification of Peskin’s Lagrangian fractional step method for the incompressible Navier-Stokes equations. This new method is substantially more efficient than the one originally proposed by Peskin. On a grid with N points, the work per time step is proportional to Nlog N. This gain in efficiency is accomplished by modifying the splitting and by using a multigrid method for the solution of the resulting systems of equations.<br><br>The method uses finite difference operators constructed with the aid of Voronoi diagrams. We have implemented it on a periodic domain in the plane. We describe an efficient algorithm for the numerical construction of periodic Voronoi diagrams in the plane. We believe that this algorithm can easily be generalized to higher space dimensions.<br><br>We report on numerical experiments with our method. We solve test problems with known solutions, and we compute a flow evolving from an initial vortex blob. Our results indicate that the method is convergent of first order.<br><br>As an application of our method, we present a fully Lagrangian variant of Peskin’s algorithm for the treatment of elastic boundaries immersed in the fluid, and we report on numerical results obtained with this algorithm.