Regular polynomial interpolation and approximation of global solutions of linear partial differential equations

We consider regular polynomial interpolation algorithms on recursively defined sets of interpolation points which approximate global solutions of arbitrary well-posed systems of linear partial differential equations. Convergence of the 'limit' of the recursively constructed family of polynomials to the solution and error estimates are obtained from a priori estimates for some standard classes of linear partial differential equations, i.e. elliptic and hyperbolic equations. Another variation of the algorithm allows to construct polynomial interpolations which preserve systems of linear partial differential equations at the interpolation points. We show how this can be applied in order to compute higher order terms of WKB-approximations of fundamental solutions of a large class of linear parabolic equations. The error estimates are sensitive to the regularity of the solution. Our method is compatible with recent developments for solution of higher dimensional partial differential equations, i.e. (adaptive) sparse grids, and weighted Monte-Carlo, and has obvious applications to mathematical finance and physics.Comment: 28 page

A New Spherical Harmonics Scheme for Multi-Dimensional Radiation Transport I: Static Matter Configurations

Recent work by McClarren & Hauck [29] suggests that the filtered spherical harmonics method represents an efficient, robust, and accurate method for radiation transport, at least in the two-dimensional (2D) case. We extend their work to the three-dimensional (3D) case and find that all of the advantages of the filtering approach identified in 2D are present also in the 3D case. We reformulate the filter operation in a way that is independent of the timestep and of the spatial discretization. We also explore different second- and fourth-order filters and find that the second-order ones yield significantly better results. Overall, our findings suggest that the filtered spherical harmonics approach represents a very promising method for 3D radiation transport calculations.Comment: 29 pages, 13 figures. Version matching the one in Journal of Computational Physic

Solving the system of radiation magnetohydrodynamics for solar physical simulations in 3d

In this study we present a finite-volumen scheme for solving the equations of radiation magnetohydrodynamics in two and three space dimensions. Among other applications this system is used to model the plasma in the solar convection zone and in the solar photosphere. It is a non--linear system of balance laws derived from the Euler equations of gas dynamics and the Maxwell equations; the energy transport through radiation is also included in the model. The starting point of our presentation is a standard explicit first and second order finite-volume scheme on both structured and unstructured grids. We first study the convergence of a finite-volume scheme applied to a scalar model problem for the full system of radiation magnetohydrodynamics. We then present modifications of the base scheme. These make it possible to approximate the system of magnetohydrodynamics with an arbitrary equation of state; they reduce errors due to a violation of the divergence constraint on the magnetic field, and they lead to an improved accuracy in the approximation of solution near an equilibrium state. These modifications significantly increase the robustness of the scheme and are essential for an accurate simulation of processes in the solar atmosphere ...thesi

Eliminating artificial boundary conditions in time-dependent density functional theory using Fourier contour deformation

We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain, FCD yields a high-order accurate numerical solution of the time-dependent Schrodinger equation directly in free space, without the need for artificial boundary conditions. Of the many existing artificial boundary condition schemes, FCD is most similar to an exact nonlocal transparent boundary condition, but it works directly on Cartesian grids in any dimension, and runs on top of the fast Fourier transform rather than fast algorithms for the application of nonlocal history integral operators. We adapt FCD to time-dependent density functional theory (TDDFT), and describe a simple algorithm to smoothly and automatically truncate long-range Coulomb-like potentials to a time-dependent constant outside of a bounded domain of interest, so that FCD can be used. This approach eliminates errors originating from the use of artificial boundary conditions, leaving only the error of the potential truncation, which is controlled and can be systematically reduced. The method enables accurate simulations of ultrastrong nonlinear electronic processes in molecular complexes in which the inteference between bound and continuum states is of paramount importance. We demonstrate results for many-electron TDDFT calculations of absorption and strong field photoelectron spectra for one and two-dimensional models, and observe a significant reduction in the size of the computational domain required to achieve high quality results, as compared with the popular method of complex absorbing potentials

Eliminating artificial boundary conditions in time-dependent density functional theory using Fourier contour deformation

We present an efficient method for propagating the time-dependent Kohn-Sham equations in free space, based on the recently introduced Fourier contour deformation (FCD) approach. For potentials which are constant outside a bounded domain, FCD yields a high-order accurate numerical solution of the time-dependent Schrodinger equation directly in free space, without the need for artificial boundary conditions. Of the many existing artificial boundary condition schemes, FCD is most similar to an exact nonlocal transparent boundary condition, but it works directly on Cartesian grids in any dimension, and runs on top of the fast Fourier transform rather than fast algorithms for the application of nonlocal history integral operators. We adapt FCD to time-dependent density functional theory (TDDFT), and describe a simple algorithm to smoothly and automatically truncate long-range Coulomb-like potentials to a time-dependent constant outside of a bounded domain of interest, so that FCD can be used. This approach eliminates errors originating from the use of artificial boundary conditions, leaving only the error of the potential truncation, which is controlled and can be systematically reduced. The method enables accurate simulations of ultrastrong nonlinear electronic processes in molecular complexes in which the inteference between bound and continuum states is of paramount importance. We demonstrate results for many-electron TDDFT calculations of absorption and strong field photoelectron spectra for one and two-dimensional models, and observe a significant reduction in the size of the computational domain required to achieve high quality results, as compared with the popular method of complex absorbing potentials

Numerical modelling of a parabolic trough solar collector

Concentrated Solar Power (CSP) technologies are gaining increasing interest in electricity generation due to the good potential for scaling up renewable energy at the utility level. Parabolic trough solar collector (PTC) is economically the most proven and advanced of the various CSP technologies. The modelling of these devices is a key aspect in the improvement of their design and performances which can represent a considerable increase of the overall efficiency of solar power plants. In the subject of modelling and improving the performances of PTCs and their heat collector elements (HCEs), the thermal, optical and aerodynamic study of the fluid flow and heat transfer is a powerful tool for optimising the solar field output and increase the solar plant performance. This thesis is focused on the implementation of a general methodology able to simulate the thermal, optical and aerodynamic behaviour of PTCs. The methodology followed for the thermal modelling of a PTC, taking into account the realistic non-uniform solar heat flux in the azimuthal direction is presented. Although ab initio, the finite volume method (FVM) for solving the radiative transfer equation was considered, it has been later discarded among other reasons due to its high computational cost and the unsuitability of the method for treating the finite angular size of the Sun. To overcome these issues, a new optical model has been proposed. The new model, which is based on both the FVMand ray tracing techniques, uses a numerical-geometrical approach for considering the optic cone. The effect of different factors, such as: incident angle, geometric concentration and rim angle, on the solar heat flux distribution is addressed. The accuracy of the new model is verified and better results than the Monte Carlo Ray Tracing (MCRT) model for the conditions under study are shown. Furthermore, the thermal behaviour of the PTC taking into account the nonuniform distribution of solar flux in the azimuthal direction is analysed. A general performance model based on an energy balance about the HCE is developed. Heat losses and thermal performances are determined and validated with Sandia Laboratories tests. The similarity between the temperature profile of both absorber and glass envelope and the solar flux distribution is also shown. In addition, the convection heat losses to the ambient and the effect of wind flow on the aerodynamic forces acting on the PTC structure are considered. To do this, detailed numerical simulations based on Large Eddy simulations (LES) are carried out. Simulations are performed at two Reynolds numbers of ReW1 = 3.6 × 105 and ReW2 = 1 × 106. These values corresponds to working conditions similar to those encountered in solar power plants for an Eurotrough PTC. The study has also considered different pitch angles mimicking the actual conditions of the PTC tracking mechanism along the day. Aerodynamic loads, i.e. drag and lift coefficients, are calculated and validatedwith measurements performed in wind tunnels. The indepen-dence of the aerodynamic coefficients with Reynolds numbers in the studied range is shown. Regarding the convection heat transfer taking place around the receiver, averaged local Nusselt number for the different pitch angles and Reynolds numbers have been computed and the influence of the parabola in the heat losses has been analysed. Last but not the least, the detailed analysis of the unsteady forces acting on the PTC structure has been conducted by means of the power spectra of several probes. The analysis has led to detect an increase of instabilities when moving the PTC to intermediate pitch angles. At these positions, the shear-layers formed at the sharp corners of the parabola interact shedding vortices with a high level of coherence. The coherent turbulence produces vibrations and stresses on the PTC structure which increase with the Reynolds number and eventually, might lead to structural failure under certain conditions