## Abstract

A system of Fredholm second kind integral equations (SKIEs) is constructed for the modified biharmonic equation in two dimensions with gradient boundary conditions. Such boundary value problem arises naturally when the incompressible Navier-Stokes equations are solved via a semi-implicit discretization scheme and the resulting boundary value problem at each time step is then solved using the pure stream-function formulation. The advantages of such an approach (Greengard and Kropinski, 1998) [14] are two fold: first, the velocity is automatically divergence free, and second, complicated (nonlocal) boundary conditions for the vorticity are avoided. Our construction, though similar to that of Farkas (1989) [10] for the biharmonic equation, is modified such that the SKIE formulation has low condition numbers for large values of the parameter. We illustrate the performance of our numerical scheme with several numerical examples. Finally, the scheme can be easily coupled with standard fast algorithms such as FFT, fast multipole methods (Greengard and Rokhlin, 1987) [15], or fast direct solvers (Ho and Greengard, 2012; Martinsson and Rokhlin, 2005) [17,25] to achieve optimal complexity, bringing accurate large-scale long-time fluid simulations within practical reach.

Original language | English (US) |
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Pages (from-to) | 113-126 |

Number of pages | 14 |

Journal | Journal of Computational Physics |

Volume | 249 |

DOIs | |

State | Published - Sep 15 2013 |

## All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Modeling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

## Keywords

- Modified biharmonic equation
- Navier-Stokes equations
- Second kind integral equation
- Unsteady Stokes flow