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

Radial basis function partition of unity operator splitting method for pricing multi-asset American options

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