Stability and error analysis for spectral Galerkin method for the Navier–Stokes equations with L2 initial data

We consider the linear time-dependent Schrodinger equation with a time- dependent smooth potential on an unbounded domain. A Galerkin spectral method with a tensor-product Hermite basis is used as a discretization in space. Discretizing the resulting ODE for the Hermite expansion coecients involves the computation of the action of the Galerkin matrix on a vector in each time step. We propose a fast algorithm for the direct computation of this matrix-vector product without actually assembling the matrix itself. The costs scale linearly in the size of the basis. Together with the application of a hyperbolically reduced basis, this reduces the computational eort considerably and helps cope with the infamous curse of dimensionality. The ap- plication of the fast algorithm is limited to the case of the potential being significantly smoother than the solution. The error analysis is based on a binary tree representation of the three-term recurrence relation for the one-dimensional Hermite functions. The fast algorithm constitutes an ecient tool for schemes involving the action of a ma- trix due to spectral discretization on a vector, and it is also applicable in the context of spectral approximations for linear problems other than the Schrodinger equation. Keywords: linear Schrodinger equation, spectral Galerkin methods, reduced index sets, fast algorithm, direct computation, curse of dimensionality, binary trees

The Lagrange–Galerkin Spectral Element Method on Unstructured Quadrilateral Grids

We present the results of a computational study of the post-processed Galerkin methods put forward by Garcia-Archilla et al. applied to the non-linear von Karman equations governing the dynamic response of a thin cylindrical panel periodically forced by a transverse point load. We spatially discretize the shell using finite differences to produce a large system of ordinary diff erential equations (ODEs). By analogy with spectral non-linear Galerkin methods we split this large system into a 'slowly' contracting subsystem and a 'quickly' contracting subsystem. We then compare the accuracy and efficiency of (i) ignoring the dynamics of the 'quick' system (analogous to a traditional spectral Galerkin truncation and sometimes referred to as 'subspace dynamics' in the finite element community when applied to numerical eigenvectors), (ii) slaving the dynamics of the quick system to the slow system during numerical integration (analogous to a non-linear Galerkin method), and (iii) ignoring the influence of the dynamics of the quick system on the evolution of the slow system until we require some output, when we 'lift' the variables from the slow system to the quick using the same slaving rule as in (ii). This corresponds to the postprocessing of Garcia-Archilla et al. We find that method (iii) produces essentially the same accuracy as method (ii) but requires only the computational power of method (i) and is thus more efficient than either. In contrast with spectral methods, this type of finite-diff erence technique can be applied to irregularly shaped domains. We feel that post-processing of this form is a valuable method that can be implemented in computational schemes for a wide variety of partial differential equations (PDEs) of practical importance.

Galerkin and subspace decomposition methods in space and time for the Navier–Stokes equations

This paper deals with developing a fast and robust numerical formulation to simulate a system of fractional PDEs. At the first stage, the time variable is approximated by a finite difference method with first-order accuracy. At the second stage, the spectral Galerkin method based upon the fractional Jacobi polynomials is employed to discretize the spatial variables. We apply a reduced-order method based upon the proper orthogonal decomposition technique to decrease the utilized computational time. The unconditional stability property and the order of convergence of the new technique are analyzed in detail. The proposed numerical technique is well known as the reduced-order spectral Galerkin scheme. Furthermore, by employing the Newton–Raphson method and semi-implicit schemes, the proposed method can be used for solving linear and nonlinear ODEs and PDEs. Finally, some examples are provided to confirm the theoretical results.

In this paper, we study an elliptic optimal control problem with $$H^1$$H1-norm state constraint. The control problem is approximated by the Galerkin spectral method, which can provide high-order accuracy and fast convergence rate. The optimality conditions and a priori error estimates are presented. A reliable a posteriori error estimator is investigated, which is helpful for developing adaptive strategy in the spectral method. Some numerical tests confirm the error estimates and illustrate the performance of the indicator.

En esta tesis se presentan varias técnicas de solución para métodos de Galerkin discontinuo (DG, p.s.s.e.i.1) de alto orden. Si bien el objetivo principal es el desarrollo de técnicas precisas y computacionalmente eficientes para resolver las ecuaciones de Navier-Stokes de flujo compresible en 3D por medio del método Discontinuous Galerkin Spectral Element Method (DGSEM), todos los métodos desarrollados en esta tesis se pueden aplicar a otras ecuaciones de conservación no lineales y a otros esquemas DG. Las técnicas de solución acá presentadas pueden ser clasificadas en dos grupos: (i) las que están orientadas a acelarar la integración temporal, y (ii) las que mejoran la discretización espacial con adaptación-p local. El primer grupo de técnicas disminuye el tiempo de cálculo dada una discretización espacial con un número fijo de grados de libertad (NDOF, p.s.s.e.i.) mediante la mejora de los algoritmos de solución. Por ese motivo, se presta especial atención al desarrollo de algoritmos eficientes para integración temporal implícita y de métodos multimalla. También se detalla una metodología para obtener el Jacobiano analítico DG para ecuaciónes de advección-difusión. Además, se demuestra que la discretización implícita del DGSEM en nodos de Gauss-Lobatto puede ser resuelta eficientemente mediante el método static condensation. Lo anterior permite resolver el problema con sistemas lineales más pequeños y mejor condicionados, lo cual disminuye los tiempos de cálculo para órdenes moderados. El segundo grupo de técnicas consiste en modificar localmente la discretización espacial para reducir NDOF sin deteriorar la precisión del método. Para lograr este obetivo, en esta tesis se desarrolla un nuevo estimador de error para el método DGSEM p-anisitrópico basado en el método _-estimation, el cual es más preciso y requiere menos recursos computacionales que enfoques previos. Posteriormente, este estimador se usa con éxito para realizar adaptación-p anisotrópica en casos estacionaros y no estacionarios, y se demuestra que éste se puede acoplar con métodos multimalla para acelerar más las simulaciones. Adicionalmente, se desarrollan dos técnicas de adaptación-p para casos no estacionarios: un algoritmo dinámico y un algoritmo estático, que es computacionalmente más eficiente en algunos casos estadísticamente estacionarios. ----------ABSTRACT---------- In this thesis, we develop space and time solution techniques for high-order Discontinuous Galerkin (DG) methods. Even though the main focus is the accurate and computationally efficient solution of the 3D compressible Navier-Stokes equations using the Discontinuous Galerkin Spectral Element Method (DGSEM), the methods developed here can be applied to other systems of nonlinear conservation laws and other DG methods. The solution techniques can be classified into two groups: (i) improved solution techniques for temporal discretizations, and (ii) local p-adaptation techniques that enhance the spatial discretization. The first group of techniques decreases the computational time for a given spatial discretization with a fixed number of degrees of freedom (NDOF) by improving the linear and nonlinear solver algorithms. In that regard, special attention is paid to the development of efficient implicit time-integration solvers and to multigrid methods (both explicit and implicit in time). We derive a methodology to obtain the analytical Jacobian of DG discretizations of nonlinear advectiondiffusion equations. Furthermore, we show that time-implicit DGSEM discretizations on Gauss-Lobatto points can be formulated as Schur complement problems and solved using the static-condensation method. This enables one to solve smaller, better-conditioned problems, which yields shorter computation times for moderate polynomial orders. The second group of techniques modifies the spatial discretization by reducing the NDOF while maintaining similar accuracy. To do that, we develop a novel truncation error estimator that uses the _-estimation method for the p-anisotropic DGSEM. The new method is more accurate and computationally cheaper than previous approaches. The error estimator is then successfully applied to perform local anisotropic p-adaptation for steady and unsteady flows. We show that the truncation error estimator can be readily coupled with multigrid techniques to obtain further speed-ups. Moreover, we develop two p-adaptation strategies for unsteady flows. One of them is a dynamic _-based anisotropic p-adaptation method. The other one is a static _-based anisotropic p-adaptation method, which may be computationally cheaper in statistically steady flows.

Exponentially accurate spectral and spectral element methods for fractional ODEs

A collocated-grid spectral difference method for compressible flows

This investigation presents a spectral method for solving the one-dimensional shallow-water wave equations. The spectral method is based on the Chebyshev collocation technique and finite-difference time stepping. The spectral method and finite-difference Preissmann scheme are applied to route a log-Pearson Type III hydrograph through a wide rectangular channel, and the results are compared. The spectral method performs better than the Preissmann scheme as long as the time-stepping errors are kept low. However, for larger time steps, the Preissmann scheme, which is almost second-order accurate in time (and second-order accurate in space) performs better than the spectral scheme, which is first-order accurate in time and has so-called infinite-order accuracy in space. This seems to indicate that the order of accuracy in time discretization is more important than that in space discretization, in numerical models, for fast-rising floods and friction-dominated flows.

Well-posedness of semilinear heat equations in L1

Implicit Runge-Kutta and spectral Galerkin methods for the two-dimensional nonlinear Riesz space fractional diffusion equation

A Petrov–Galerkin spectral element method for fractional elliptic problems

The discontinuous Galerkin Spectral Element Method (DG-SEM) is highly attractive for both DNS and LES of turbulent flows due to its low dispersion and dissipation errors, but also because of its good parallel scaling property. We show that especially for underresolved simulations the method has highly beneficial properties for LES, as well as for the direct treatment of the acoustic propagation. We also discuss different approaches for non-reflecting boundary conditions for DG-SEM and show the behavior of the methods for airfoil flows. Our main intend is to directly simulate trailing edge noise for airfoil flows at medium Reynolds numbers.

We describe an efficient parallelization strategy for the discontinuous Galerkin spectral element method, illustrated by a structured grid framework. Target applications are large scale DNS and LES calculations on massively parallel systems. Due to the simple and efficient formulation of the method, a parallelization aiming at one-element-per-processor calculations is feasible; a highly desired feature for emerging multi- and many-core architectures. We show scale-up tests on up to 131,000 processors.KeywordsDiscontinuous GalerkinSpectral ElementMPIParallelHPC