Optimal multistage relaxation with automatic parameter selection

This research developed a numerical method that solves complicated fluid flow problems without requiring end-user expertise with the solver. This method is capable of obtaining a spatially accurate solution in the same time or better as a skilled user with a conventional solver. An explicit preconditioned multigrid solver was used in this research with a multistage relaxation method. The prosposed method utilizies a database with optimized relaxation method parameters for different local flow and mesh conditions. The parameters are optimized for the relaxation such that the error modes in a complex Fourier series expansion of the residual can be quickly reduced. The convergence time and iteration count of this method was compared against the same solver using default input values, as well as a pre-optimized solver, to simulate a skilled user for various geometries. Improvements in both comparisons were demonstrated.

Optimal multistage relaxation with automatic parameter selection

This research developed a numerical method that solves complicated fluid flow problems without requiring end-user expertise with the solver. This method is capable of obtaining a spatially accurate solution in the same time or better as a skilled user with a conventional solver. An explicit preconditioned multigrid solver was used in this research with a multistage relaxation method. The prosposed method utilizies a database with optimized relaxation method parameters for different local flow and mesh conditions. The parameters are optimized for the relaxation such that the error modes in a complex Fourier series expansion of the residual can be quickly reduced. The convergence time and iteration count of this method was compared against the same solver using default input values, as well as a pre-optimized solver, to simulate a skilled user for various geometries. Improvements in both comparisons were demonstrated.

Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the ruin probability and analyze its convergence. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.

Effect of porous layer on the Faraday instability in viscous liquid

A linear stability analysis of a viscous liquid on a vertically oscillating porous plane is performed for infinitesimal disturbances of arbitrary wavenumbers. A time-dependent boundary value problem is derived and solved based on the Floquet theory along with the complex Fourier series expansion. Numerical results show that the Faraday instability is dominated by the subharmonic solution at high forcing frequency, but it responds harmonically at low forcing frequency. The unstable regions corresponding to both subharmonic and harmonic solutions enhance with the increasing value of permeability and yields a destabilizing effect on the Faraday instability. Further, the presence of porous layer makes faster the transition process from subharmonic instability to harmonic instability in the wavenumber regime. In addition, the first harmonic solution shrinks gradually and becomes an unstable island, and ultimately disappears from the neutral curve if the porous layer thickness is increased. In contrast, the first and second subharmonic solutions coalesce, and the onset of Faraday instability is dominated by the subharmonic solution. In a special case, the study of Faraday instability of a viscous liquid on a porous substrate can be replaced by a study of Faraday instability of a viscous liquid on a slippery substrate when the permeability of the porous substrate is very low. Further, the Faraday instability can be destabilized by introducing a slip effect at the bottom plane.