A fast algorithm for particle simulations
Journal of Computational Physics
Laplace's equation and the Dirichlet-Neumann map in multiply connected domains
Journal of Computational Physics
Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
Integral equation methods for Stokes flow and isotropic elasticity in the plane
Journal of Computational Physics
High-Order Corrected Trapezoidal Quadrature Rules for Singular Functions
SIAM Journal on Numerical Analysis
An Integral Equation Approach to the Incompressible Navier--Stokes Equations in Two Dimensions
SIAM Journal on Scientific Computing
Hybrid Gauss-Trapezoidal Quadrature Rules
SIAM Journal on Scientific Computing
Spectral methods in MatLab
A Generalized Fast Multipole Method for Nonoscillatory Kernels
SIAM Journal on Scientific Computing
A New Fast-Multipole Accelerated Poisson Solver in Two Dimensions
SIAM Journal on Scientific Computing
A kernel-independent adaptive fast multipole algorithm in two and three dimensions
Journal of Computational Physics
A fast direct solver for boundary integral equations in two dimensions
Journal of Computational Physics
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Accelerating the convergence of spectral deferred correction methods
Journal of Computational Physics
An adaptive fast solver for the modified Helmholtz equation in two dimensions
Journal of Computational Physics
Short note: A kernel independent fast multipole algorithm for radial basis functions
Journal of Computational Physics
An Accelerated Kernel-Independent Fast Multipole Method in One Dimension
SIAM Journal on Scientific Computing
The black-box fast multipole method
Journal of Computational Physics
Fast integral equation methods for the modified Helmholtz equation
Journal of Computational Physics
Integral-equation-based fast algorithms and graph-theoretic methods for large-scale simulations
Integral-equation-based fast algorithms and graph-theoretic methods for large-scale simulations
Journal of Computational Physics
Hi-index | 31.45 |
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.