A Kernel-Independent Treecode Based on Barycentric Lagrange Interpolation

Abstract

A kernel-independent treecode (KITC) is presented for fast summation of particle interactions. The method employs barycentric Lagrange interpolation at Chebyshev points to approximate well-separated particle-cluster interactions. The KITC requires only kernel evaluations, is suitable for non-oscillatory kernels, and relies on the scale-invariance property of barycentric Lagrange interpolation. For a given level of accuracy, the treecode reduces the operation count for pairwise interactions from O(N 2 ) to O(Nlog N), where N is the number of particles in the system. The algorithm is demonstrated for systems of regularized Stokeslets and rotlets in 3D, and numerical results show the treecode performance in terms of error, CPU time, and memory consumption. The KITC is a relatively simple algorithm with low memory consumption, and this enables a straightforward OpenMP parallelization.

A domain decomposition solution of the Stokes-Darcy system in 3D based on boundary integrals

“…Another challenge in such a simulation is the high computational cost of the direct evaluation of the resulting dense linear systems, which is O(N 2 ), where N is the number of unknowns. To be able to simulate large enough systems, the computational cost will be reduced using fast summation techniques such as the kernel-independent treecode [58] or a fast multipole method [56].…”

“…Note that the discretized integrals were implemented to take advantage of the target-source symmetry in all kernels, reducing the computational time in half. If very large systems are considered, the computational time could be reduced to O(N log N ) or O(N ) using a fast summation algorithm such as a treecode [58] or a fast multipole method [56].…”

The Role of the Double Layer Potential in Regularized Stokeslet Models of Self-Propulsion

“…Other key areas of methodological development for the method of regularized stokeslets include development of image systems for plane boundaries [13,45,46]; extension to Brinkman/oscillatory Stokes flow [47], triply [48], doubly [49,50] and singly [51] periodic boundary conditions; the use of radial basis functions to represent force distributions [52]; improvement to the near-field regularization error [53], far-field regularization error [54]; Richardson extrapolation in regularization parameter [55]; and perhaps most powerfully, methods based on kernel-independent fast multipole method [56,57] and treecode [58]. Regularization at the level of the boundary integral equation itself combined with asymptotic corrections (a related but different perspective from regularization of the equation from which the fundamental solution is derived) has been shown to provide highly accurate results for problems including near-contact of droplets without the need for specialized quadrature [59,60].…”

The Role of the Double-Layer Potential in Regularised Stokeslet Models of Self-Propulsion

“…Other key areas of methodological development for the method of regularised stokeslets include the development of image systems for plane boundaries [13,45,46]; extension to Brinkman/oscillatory Stokes flow [47], triply [48], doubly [49,50] and singly [51] periodic boundary conditions; the use of radial basis functions to represent force distributions [52]; improvement to the near-field regularisation error [53], far-field regularisation error [54]; Richardson extrapolation in the regularisation parameter [55]; and perhaps most powerfully, methods based on kernel-independent fast multipole method [56,57] and treecode [58]. Regularisation at the level of the boundary integral equation itself combined with asymptotic corrections (a related but different perspective from regularisation of the equation from which the fundamental solution is derived) has been shown to provide highly accurate results for problems including the near contact of droplets without the need for specialised quadrature [59,60].…”