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Register a new userNumerical Simulation of Microflows using Hermite Spectral Methods
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We propose a Hermite spectral method for the spatially inhomogeneous Boltzmann equation. For the inverse-power-law model, we generalize an approximate quadratic collision operator defined in the normalized and dimensionless setting to an operator for arbitrary distribution functions. An efficient algorithm with a fast transform is introduced to discretize this new collision operator. The method is tested for one-dimensional benchmark microflow problems.
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We extend the proof in [M.~Crouzeix and C.~Palencia, {em The numerical range is a $(1 + sqrt{2})$-spectral set}, SIAM Jour.~Matrix Anal.~Appl., 38 (2017), pp.~649-655] to show that other regions in the complex plane are $K$-spectral sets. In particular, we show that various annular regions are $(1 + sqrt{2} )$-spectral sets and that a more general convex region with a circular hole or cutout is a $(3 + 2 sqrt{3} )$-spectral set. We demonstrate how these results can be used to give bounds on the convergence rate of the GMRES algorithm for solving linear systems and on that of rational Krylov subspace methods for approximating $f(A)b$, where $A$ is a square matrix, $b$ is a given vector, and $f$ is a function that can be uniformly approximated on such a region by rational functions with poles outside the region.
Global spectral analysis (GSA) is used as a tool to test the accuracy of numerical methods with the help of canonical problems of convection and convection-diffusion equation which admit exact solutions. Similarly, events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with theoretical results of homogeneous isotropic turbulence due to Kolmogorov, as given in Turbulence -U. Frisch, Cambridge Univ. Press, UK (1995). However, numerical methods for the simulation of this problem are not calibrated, as by using GSA of convection and/or convection-diffusion equation. This is with the exception in A critical assessment of simulations for transitional and turbulence flows-Sengupta, T.K., In Proc. of IUTAM Symp. on Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, pp 491-532, World Sci. Publ. Co. Pte. Ltd., Singapore (2016), where such a calibration has been advocated with the help of convection equation. For turbulent flows, an extreme event is characterized by the presence of length scales smaller than the Kolmogorov length scale, a heuristic limit for the largest wavenumber present without being converted to heat. With growing computer power, recently many simulations have been reported using a pseudo-spectral method, with spatial discretization performed in Fourier spectral space and a two-stage, Runge-Kutta (RK2) method for time discretization. But no analyses are reported to ensure high accuracy of such simulations. Here, an analysis is reported for few multi-stage Runge-Kutta methods in the Fourier spectral framework for convection and convection-diffusion equations. We identify the major source of error for the RK2-Fourier spectral method using GSA and also show how to avoid this error and specify numerical parameters for achieving highest accuracy possible to capture extreme events in turbulent flows.
We discuss a new numerical schema for solving the initial value problem for the Korteweg-de Vries equation for large times. Our approach is based upon the Inverse Scattering Transform that reduces the problem to calculating the reflection coefficient of the corresponding Schrodinger equation. Using a step-like approximation of the initial profile and a fragmentation principle for the scattering data, we obtain an explicit recursion formula for computing the reflection coefficient, yielding a high resolution KdV solver. We also discuss some generalizations of this algorithm and how it might be improved by using Haar and other wavelets.
Graphics Processing Unit (GPU) computing is becoming an alternate computing platform for numerical simulations. However, it is not clear which numerical scheme will provide the highest computational efficiency for different types of problems. To this end, numerical accuracies and computational work of several numerical methods are compared using a GPU computing implementation. The Correction Procedure via Reconstruction (CPR), Discontinuous Galerkin (DG), Nodal Discontinuous Galerkin (NDG), Spectral Difference (SD), and Finite Volume (FV) methods are investigated using various reconstruction orders. Both smooth and discontinuous cases are considered for two-dimensional simulations. For discontinuous problems, MUSCL schemes are employed with FV, while CPR, DG, NDG, and SD use slope limiting. The computation time to reach a set error criteria and total time to complete solutions are compared across the methods. It is shown that while FV methods can produce solutions with low computational times, they produce larger errors than high-order methods for smooth problems at the same order of accuracy. For discontinuous problems, the methods show good agreement with one another in terms of solution profiles, and the total computational times between FV, CPR, and SD are comparable.
In the paper, the pricing of Quanto options is studied, where the underlying foreign asset and the exchange rate are correlated with each other. Firstly, we adopt Bayesian methods to estimate unknown parameters entering the pricing formula of Quanto options, including the volatility of stock, the volatility of exchange rate and the correlation. Secondly, we compute and predict prices of different four types of Quanto options based on Bayesian posterior prediction techniques and Monte Carlo methods. Finally, we provide numerical simulations to demonstrate the advantage of Bayesian method used in this paper comparing with some other existing methods. This paper is a new application of the Bayesian methods in the pricing of multi-asset options.
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