Spectral methods for CFD

One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched

An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry

We present a numerical method for a class of boundary value problems on the unit interval which feature a type of powerlaw nonlinearity. In order to numerically solve this type of nonlinear boundary value problems, we construct a kind of spectral collocation method. The spatial approximation is based on shifted Jacobi polynomials J(n)((alpha,beta))(r) with alpha,beta epsilon (-1, infinity), r epsilon (0,1) and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes for the spectral method. After deriving the method for a rather general class of equations, we apply it to several specific examples. One natural example is a nonlinear boundary value problem related to the Yamabe problem which arises in mathematical physics and geometry. A number of specific numerical experiments demonstrate the accuracy and the efficiency of the spectral method. We discuss the extension of the method to account for more complicated forms of nonlinearity

Numerical Methods for Neutron Transport Calculations of Nuclear Reactors

The objective of this thesis, which in clearly inspired by an industrial framework, is to try and narrow the gap between theoretical neutron modelling and application in the context of nuclear reactor design. This thesis is divided into three main chapters, preceded by a general overview. This structure reflects the three main topics which were chosen for this research project. The first topic develops the Spectral Element Method (SEM) approach and its use in conjunction with transport approximations. As it is documented in the specialized numerical analysis handbooks and in previous works by the author, the method has an excellent convergence rate which outperforms most classical schemes, but it has also some important drawbacks which sometimes seem to discourage its use for linear transport problems applied to nontrivial benchmarks. In order to elaborate the methodology of the specific problems encountered in reactor physics, three aspects are addressed looking for improvements. The first topic analyzed is related to the convergence order, whose value is less straightforward to define a priori by means of functional analysis than other numerical schemes. The adjective “spectral” refers in fact to the maximum order claimed, exponential with respect to the average size of the mesh. A comprehensive set of convergence tests is carried out applying SEM to a few transport models and with the aid of manufactured solutions, thus isolating the numerical effects from the deviations which are due only to modelling approximations. The hypothesis of grid conformity is also relaxed, replacing the classical Galerkin variational formulation with the Discontinuous Galerkin theory, characterized by a more flexible treatment of the mesh interfaces; this scheme allows local grid refinement and opens the way, in perspective, to mesh adaption. Finally, a simple and sufficiently flexible technique to deform the boundaries of each mesh is introduced and applied, in order to adapt the grid to curved geometries. In this way, the advantages of SEM can be applied to a vast class of common problems like lattice calculations. Moreover, thanks to a change of the basis functions used in SEM, it is possible to obtain elements with three sides (straight or deformed), that are a typical war horse of the Finite Element approach. The second topic is essentially devoted to the most “industrial” part of the thesis, developed entirely during the stay of the author in the AREVA NP headquarters in Paris. In AREVA, and in all other nuclear engineering enterprises, neutron diffusion is still the preferred neutronic model for full-core studies. Better approximations are reserved for library preparation, fuel studies and code validation, none of these being typically too much time or budget-constrained. Today needs start to require a certain level of improvement also in full-core analyses, trying to fitly model localized dis-homogeneities and reduce the penalizing engineering margins which are taken as provisions. On the other hand, a change in the model does not mean only an effort to write a new code, but has huge follow-ups due to the validation processes required by the authorities. Second-order transport may support the foreseen methodology update because it can be implemented re-using diffusion routines as the computational engine. The AN method, a second-order approximation of the transport equation, has been introduced in some studies, and its effect is discussed. Moreover, some effort has been reserved to the introduction of linear anisotropy in the model. The last topic deals with ray effects; they are a known issue of the discrete ordinate approach (SN methods) which is responsible for a reduction in the accuracy of the solution, especially in penetration problems with low scattering, like several shielding calculations performed for operator safety concerns. Ray effects are here characterized from a formal point of view in both static and time dependent situations. Then, quantitative indicators are defined to help with the interpretation of the SN results. Based on these studies, some mitigation measures are proposed and their efficacy is discussed

Fast Spectral Collocation Method for Solving Nonlinear Time-Delayed Burgers-Type Equations with Positive Power Terms

Since the collocation method approximates ordinary differential equations, partial differential equations, and integral equations in physical space, it is very easy to implement and adapt to various problems, including variable coefficient and nonlinear differential equations. In this paper, we derive a Jacobi-Gauss-Lobatto collocation method (J-GL-C) to solve numerically nonlinear time-delayed Burgers-type equations. The proposed technique is implemented in two successive steps. In the first one, we apply nodes of the Jacobi-Gauss-Lobatto quadrature which depend upon the two general parameters , and the resulting equations together with the two-point boundary conditions constitute a system of ordinary differential equations (ODEs) in time. In the second step, the implicit Runge-Kutta method of fourth order is applied to solve a system of ODEs of second order in time. We present numerical results which illustrate the accuracy and flexibility of these algorithms

Efficient Methods for Multidimensional Global Polynomial Approximation with Applications to Random PDEs

In this work, we consider several ways to overcome the challenges associated with polynomial approximation and integration of smooth functions depending on a large number of inputs. We are motivated by the problem of forward uncertainty quantification (UQ), whereby inputs to mathematical models are considered as random variables. With limited resources, finding more efficient and accurate ways to approximate the multidimensional solution to the UQ problem is of crucial importance, due to the “curse of dimensionality” and the cost of solving the underlying deterministic problem. The first way we overcome the complexity issue is by exploiting the structure of the approximation schemes used to solve the random partial differential equations (PDE), thereby significantly reducing the overall cost of the approximation. We do this first using multilevel approximations in the physical variables, and second by exploiting the hierarchy of nested sparse grids in the random parameter space. With these algorithmic advances, we provably decrease the complexity of collocation methods for solving random PDE problems. The second major theme in this work is the choice of efficient points for multidimensional interpolation and interpolatory quadrature. A major consideration in interpolation in multiple dimensions is the balance between stability, i.e., the Lebesgue constant of the interpolant, and the granularity of the approximation, e.g., the ability to choose an arbitrary number of interpolation points or to adaptively refine the grid. For these reasons, the Leja points are a popular choice for approximation on both bounded and unbounded domains. Mirroring the best-known results for interpolation on compact domains, we show that Leja points, defined for weighted interpolation on R, have a Lebesgue constant which grows subexponentially in the number of interpolation nodes. Regarding multidimensional quadratures, we show how certain new rules, generated from conformal mappings of classical interpolatory rules, can be used to increase the efficiency in approximating multidimensional integrals. Specifically, we show that the convergence rate for the novel mapped sparse grid interpolatory quadratures is improved by a factor that is exponential in the dimension of the underlying integral

A Modified Generalized Laguerre-Gauss Collocation Method for Fractional Neutral Functional-Differential Equations on the Half-Line

The modified generalized Laguerre-Gauss collocation (MGLC) method is applied to obtain an approximate solution of fractional neutral functional-differential equations with proportional delays on the half-line. The proposed technique is based on modified generalized Laguerre polynomials and Gauss quadrature integration of such polynomials. The main advantage of the present method is to reduce the solution of fractional neutral functional-differential equations into a system of algebraic equations. Reasonable numerical results are achieved by choosing few modified generalized Laguerre-Gauss collocation points. Numerical results demonstrate the accuracy, efficiency, and versatility of the proposed method on the half-line