A radial basis function partition of unity collocation method for convection-diffusion equations ⋆

Abstract Numerical solution of multi-dimensional PDEs is a challenging problem with respect to computational cost and memory requirements, as well as regarding representation of realistic geometries and adaption to solution features. Meshfree methods such as global radial basis function approximation have been successfully applied to several types of problems. However, due to the dense linear systems that need to be solved, the computational cost grows rapidly with dimension. In this paper, we instead propose to use a locally supported RBF collocation method based on a partition of unity approach to numerically solve time-dependent PDEs. We investigate the stability and accuracy of the method for convection-diffusion problems in two space dimensions as well as for an American option pricing problem. The numerical experiments show that we can achieve both spectral and high-order algebraic convergence for convection-diffusion problems, and that we can reduce the computational cost for the option pricing problem by adapting the node layout to the problem characteristics

Numerical solution of the viscous flow past a cylinder with a non-global yet spectrally convergent meshless collocation method

Proceeding of: 11th International Conference on Spectral and High-Order Methods (ICOSAHOM'16), June 27-July 1, 2016, Rio de Janeiro, Brazil.The flow of a viscous fluid past a cylinder is a classical problem in fluid-structure interaction and a benchmark for numerical methods in computational fluid dynamics. We solve it with the recently introduced radial basis function-based partition of unity method (RBF-PUM), which is a spectrally convergent collocation meshless scheme well suited to this kind of problem. The resulting discrete system of nonlinear equations is tackled with a trust-region algorithm, whose performance is much enhanced by the analytic Jacobian which is provided alongside. Preliminary results up to Re = 60 with just 1292 nodes are shown.F. Bernal acknowledges support from FCT grant SFRH/BPD/79986/2011 and INESC-ID. A. Heryudono is partially supported by NSF Grant DMS 1552238Publicad

Adaptive residual subsampling methods for radial basis function interpolation and collocation problems

We construct a new adaptive algorithm for radial basis func-tions (RBFs) method applied to interpolation, boundary-value, and initial-boundary-value problems with localized features. Nodes can be added and removed based on residuals evaluated at a finer point set. We also adapt the shape parameters of RBFs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix. The performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains

Single-Equation Models for the Tear Film in a Blink Cycle: Realistic Lid Motion

We consider model problems for the tear film over multiple blink cycles that utilize a single equation for the tear film; the single nonlinear partial differential equation (PDE) that governs the film thickness arises from lubrication theory. The two models that we consider arise from considering the absence of naturally occuring surfactant and the case when the surfactant is strongly affecting the surface tension. The film is considered on a time-varying domain length with specified film thickness and volume flux at each end; only one end of the domain is moving, which is analogous to the upper eyelid moving with each blink. Realistic lid motion from observed blinks is included in the model with end fluxes specified to more closely match the blink cycle than those previously reported. Numerical computations show quantitative agreement with in vivo tear film thickness measurements under partial blink conditions. A transition between periodic and nonperiodic solutions has been estimated as a function of closure fraction and this may be a criterion for what is effectively a full blink according to fluid dynamics. 1