We present in this paper a unified framework for analyzing the spectral methods in unbounded domains using mapped Jacobi, Laguerre and Hermite functions. A detailed comparison of the convergence rates of these spectral methods for solutions with typical decay behaviors is carried out, both theoretically and computationally. A brief review on some of the recent advances in the spectral methods for unbounded domains is also presented.

This paper presents a review of the current state-of-the-art of numerical methods for stochastic computations. The focus is on efficient high-order methods suitable for practical applications, with a particular emphasis on those based on generalized polynomial chaos (gPC) methodology. The framework of gPC is reviewed, along with its Galerkin and collocation approaches for solving stochastic equations. Properties of these methods are summarized by using results from literature. This paper also attempts to present the gPC based methods in a unified framework based on an extension of the classical spectral methods into multi-dimensional random spaces.

We present a coupled thermal-fluid model for Bénard-Marangoni convection in a three-dimensional fluid layer. The governing equations are derived in detail for two reasons: first, we do not assume a flat free surface as commonly done; and second, we prepare for the use of flexible discretizations. The governing equations are discretized using spectral elements in space and an operator splitting approach in time. Since we are here primarily interested in steady state solutions, the focus is on the spatial discretization. The overall computational approach is very attractive to use for several reasons: (i) the solution can be expected to have a high degree of regularity, and rapid convergence can be expected; (ii) the spectral element decomposition automatically gives a convenient parameterization of the free surface that allows powerful results from differential geometry to easily be exploited; (iii) free surface deformation can readily be included; (iv) both normal and tangential stresses are conveniently accounted for through a single surface integral; (v) no differentiation of the surface tension is necessary in order to include thermocapillary effects (due to integration-by-parts twice); (vi) the geometry representation of the free surface need only be C⁰ across element boundaries even though curvature effects are included. Three-dimensional simulation results are presented, including the free surface deflection due to buoyancy and thermocapillary effects.

This paper is concerned with preconditioners for interior penalty discontinuous Galerkin discretizations of second-order elliptic boundary value problems. We extend earlier related results in [7] in the following sense. Several concrete realizations of splitting the nonconforming trial spaces into a conforming and (remaining) nonconforming part are identified and shown to give rise to uniformly bounded condition numbers. These asymptotic results are complemented by numerical tests that shed some light on their respective quantitative behavior. 

This paper designs a segmentation method for an image based on its Fourier spectral data. An edge map is generated directly from the Fourier coefficients without first reconstructing the image in pixelated form. Consequently the internal boundaries of the edge map are not blurred by any (filtered) Fourier reconstruction. The edge map is then processed with an edge linking segmentation algorithm. We include examples from magnetic resonance imaging (MRI). Our results illustrate some potential benefits of using high order methods in imaging.

In this paper, we propose a composite Laguerre-Legendre pseudospectral method for exterior problems with a square obstacle. Some results on the composite Laguerre-Legendre interpolation, which is a set of piecewise mixed interpolations coupled with domain decomposition, are established. As examples of applications, the composite pseudospectral schemes are provided for two model problems. The convergence of proposed schemes are proved. Efficient algorithms are implemented. Numerical results demonstrate the spectral accuracy in space of this new approach.

In this paper, we reformulate the piecewise linear discontinuous Galerkin (DG) method for solving two dimensional steady state scalar conservation laws in the framework of residual distribution (RD) schemes. This allows us to propose a new class of nonlinear stabilization that does not destroy the formal accuracy of the schemes. Numerical results are shown to demonstrate the behavior of this approach. 

This paper describes an iterative solution technique for partial differential equations involving the grad(div) operator, based on a domain decomposition. Iterations are performed to solve the solution on the interface. We identify the transmission relationships through the interface. We relate the approach to a Steklov-Poincaré operator, and we illustrate the performance of technique through some numerical experiments. 

We extend the construction and analysis of the non-overlapping Schwarz preconditioners proposed in [2,3] to the (non-consistent) super penalty discontinuous Galerkin methods introduced in [5] and [8]. We show that the resulting preconditioners are scalable, and we provide the convergence estimates. We also present numerical experiments confirming the sharpness of the theoretical results.

We present the application of the recent physics-conforming COOL method [2, 4] to the eigenvalue problem of a cylindrical waveguide filled with unmagnetized plasma. Using the Fourier transform only along the waveguide and not in poloidal direction, this is a relevant test case for a numerical discretization method in two dimensions (radial and poloidal). Analytically, the frequency spectrum consists of discrete electromagnetic parts and, depending on the electron density profile of the plasma, of infinitely degenerate and/or continuous, essentially electrostatic parts. If the plasma is absent, the latter reduces to the infinitely degenerate zero eigenvalue of electrostatics. A good discretization method for the Maxwell equations must reproduce these properties. It is shown here that the COOL method meets this demand properly and to very high precision.

New direct spectral solvers for the 3D Helmholtz equation in a finite cylindrical region are presented. A purely variational (no collocation) formulation of the problem is adopted, based on Fourier series expansion of the angular dependence and Legendre polynomials for the axial dependence. A new Jacobi basis is proposed for the radial direction overcoming the main disadvantages of previously developed bases for the Dirichlet problem. Nonhomogeneous Dirichlet boundary conditions are enforced by a discrete lifting and the vector problem is solved by means of a classical uncoupling technique. In the considered formulation, boundary conditions on the axis of the cylindrical domain are never mentioned, by construction. The solution algorithms for the scalar equations are based on double diagonalization along the radial and axial directions. The spectral accuracy of the proposed algorithms is verified by numerical tests.

We consider high order methods for the one-dimensional Helmholtz equation and frequency-Maxwell system. We demand that the scheme be higher order even when the coefficients are discontinuous. We discuss the connection between schemes for the second-order scalar Helmholtz equation and the first-order system for the electromagnetic or acoustic applications.

The finite volume (FV) method is the dominating discretization technique for computational fluid dynamics (CFD), particularly in the case of compressible fluids. The discontinuous Galerkin (DG) method has emerged as a promising high-accuracy alternative. The standard DG method reduces to a cell-centered FV method at lowest order. However, many of today’s CFD codes use a vertex-centered FV method in which the data structures are edge based. We develop a new DG method that reduces to the vertex-centered FV method at lowest order, and examine here the new scheme for scalar hyperbolic problems. Numerically, the method shows optimal-order accuracy for a smooth linear problem. By applying a basic hp-adaption strategy, the method successfully handles shocks. We also discuss how to extend the FV edge-based data structure to support the new scheme. In this way, it will in principle be possible to extend an existing code employing the vertex-centered and edge-based FV discretization to encompass higher accuracy through the new DG method.

Variations on space-time Discontinuous Galerkin (STDG) discretization associated to Runge-Kutta schemes are developed. These new schemes while keeping the original scheme order can improve accuracy and stability. Numerical analysis is made on academic test cases and efficiency of these schemes is shown on propagating pressure waves. 

Approximating a function from its values f(x_i) at a set of evenly spaced points x_i through (N+1)-point polynomial interpolation often fails because of divergence near the endpoints, the "Runge Phenomenon". Here we briefly describe seven strategies, each employing a single polynomial over the entire interval, to wholly or partially defeat the Runge Phenomenon such that the error decreases exponentially fast with N. Each is successful in obtaining high accuracy for Runge's original example. Unfortunately, each of these single-interval strategies also has liabilities including, depending on the method, various permutations of inefficiency, ill-conditioning and a lack of theory. Even so, the Fourier Extension and Gaussian RBF methods are worthy of further development. 

We extend the results on minimal stabilization of Burmanand Stamm [J. Sci. Comp., 33 (2007), pp. 183-208] to the case of the local discontinuous Galerkin methods on mixed form. The penalization term on the faces is relaxed to act only on a part of the polynomial spectrum. Stability in the form of a discrete inf-sup condition is proved and optimal convergence follows. Some numerical examples using high order approximation spaces illustrate the theory. 

An application of recent uncertainty quantification techniques to Wind Engineering is presented. In particular, the study of the effects of small geometric changes in the Sunshine Skyway Bridge deck on its aerodynamic behavior is addressed. This results in the numerical solution of a proper PDE posed in a domain affected by randomness, which is handled through a mapping approach. A non-intrusive Polynomial Chaos expansion allows to transform the stochastic problem into a deterministic one, in which a commercial code is used as a black-box for the solution of a number of Reynolds-Averaged Navier-Stokes simulations. The use of proper Gauss-Patterson nested quadrature formulas with respect to a Truncated Weibull probability density function permits to limit the number of these computationally expensive simulations, though maintaining a sufficient accuracy. Polynomial Chaos approximations, statistical moments and probability density functions of time-independent quantities of interest for the engineering applications are obtained.

In this work we present a family of relaxation schemes for nonlinear convection diffusion problems, which can tackle also the cases of degenerate diffusion and of convection dominated regimes. The schemes proposed can achieve any order of accuracy, give non-oscillatory solutions even in the presence of singularities and their structure depends only weakly on the particular PDE being integrated. One and two dimensional results are shown, and a nonlinear stability estimate is given.

We propose an adaptive strategy for solving high frequency Helmholtz scattering problems. The method is based on the uniaxial PML method to truncate the scattering problem which is defined in the unbounded domain into the bounded domain. The parameters in the uniaxial PML method are determined by sharp a posteriori error estimates developed by Chen and Wu [8]. An hp-adaptive finite element strategy is proposed to solve the uniaxial PML equation. Numerical experiments are included which indicate the desirable exponential decay property of the error.

We compute multivalued solutions of one- and two-dimensional pressureless gas dynamics equations by deterministic particle methods. Point values of the computed solutions are to be recovered from their singular particle approximations using some smoothing procedure. We study several recovery strategies and demonstrate ability of the particle methods to achieve high resolution. 

In recent years, Fourier spectral methods have emerged as competitive numerical methods for large-scale phase field simulations of microstructures in computational materials sciences. To further improve their effectiveness, we recently developed a new adaptive Fourier-spectral semi-implicit method (AFSIM) for solving the phase field equation by combining an adaptive moving mesh method and the semi-implicit Fourier spectral algorithm. In this paper, we present the application of AFSIM to the Cahn-Hilliard equation with inhomogeneous, anisotropic elasticity. Numerical implementations and test examples in both two and three dimensions are considered with a particular illustration using the well-studied example of mis-fitting particles in a solid as they approach to their equilibrium shapes. It is shown that significant savings in memory and computational time is achieved while accurate solutions are preserved.

In this paper, we consider high order multi-domain penalty spectral Galerkin methods for the approximation of hyperbolic conservation laws. This formulation has a penalty parameter which can vary in space and time, allowing for flexibility in the penalty formulation. This flexibility is particularly advantageous for problems with an inhomogeneous mesh. We show that the discontinuous Galerkin method is equivalent to the multi-domain spectral penalty Galerkin method with a particular value of the penalty parameter. The penalty parameter has an effect on both the accuracy and stability of the method. We examine the numerical issues which arise in the implementation of high order multi-domain penalty spectral Galerkin methods. The coefficient truncation method is proposed to prevent the rapid error growth due to round-off errors when high order polynomials are used. Finally, we show that an inconsistent evaluation of the integrals in the penalty method may lead to growth of errors. Numerical examples for linear and nonlinear problems are presented.

A fluid mixture model of tissue deformations has been studied in this paper. The model is a mixed system of nonlinear hyperbolic and elliptic partial differential equations. Both theoretical linear stability and numerical analysis are presented. Comparisons between standard numerical methods that utilize Runge-Kutta methods coupled with the WENO scheme and the immersed interface methods are given. Numerical examples are also presented.

The paper addresses the Large-Eddy Simulation (LES) of the turbulent wake of the Ahmed car model. To this end we use a Fourier-Chebyshev multi-domain solver and the LES capability is implemented through the use of the Spectral Vanishing Viscosity (SVV) method, completed with a near-wall correction. A "pseudo-penalization" technique is used to model the bluff body. Comparisons of the present SVV-LES results with the experiments and also with a more classical Finite Volume LES are provided. 

In this paper a Chebyshev collocation method is used for solving numerically an optimal boundary control problem in a thermoconvective fluid flow. The aim of this study is to demonstrate the capabilities of these numerical techniques for handling this kind of problems. As the problem is treated in the primitive variable formulation additional boundary conditions for the pressure and the auxiliary pressure fields are required to avoid spurious modes. A dependence of the convergence of the method on the penalizing parameter that appears in the functional cost is observed. As this parameter approaches zero some singular behaviour in the control function is observed and the order of the method decreases. These singularities are irrelevant in the problem as a regularized control function produces the same results.

An efficient p-multigrid method is developed to solve the algebraic systems which result from the approximation of elliptic problems with the so-called Fekete-Gauss Spectral Element Method, which makes use of the Fekete points of the triangle as interpolation points and of the Gauss points as quadrature points. A multigrid strategy is defined by comparison of different prolongation/restriction operators and coarse grid algebraic systems. The efficiency and robustness of the approach, with respect to the type of boundary condition and to the structured/unstructured nature of the mesh, are highlighted through numerical examples.

The present paper introduces bilinear forms that are equivalent to the recovery-based discontinuous Galerkin formulation introduced by Van Leer in 2005. The recovery method approximates the solution of the diffusion equation in a discontinuous function space, while inter-element coupling is achieved by a local L₂ projection that recovers a smooth continuous function underlying the discontinuous approximation. Here we introduce the concept of a local “recovery polynomial basis” – smooth polynomials that are in the weak sense indistinguishable from the discontinuous basis polynomials – and show it allows us to eliminate the recovery procedure. The recovery method reproduces the symmetric discontinuous Galerkin formulation with additional penalty-like terms depending on the targeted accuracy of the method. We present the unique link between the recovery method and discontinuous Galerkin bilinear forms. 

Detection of edges in piecewise smooth functions is important in many applications. Higher order reconstruction algorithms in image processing and post processing of numerical solutions to partial differential equations require the identification of smooth domains, creating the need for algorithms that will accurately identify discontinuities in a given function as well as those in its gradient. This work expands the use of the polynomial annihilation edge detector, (Archibald, Gelb and Yoon, 2005), to locate discontinuities in the gradient given irregularly sampled point values of a continuous function. The idea is to preprocess the given data by calculating the derivative, and then to use the polynomial annihilation edge detector to locate the jumps in the derivative. We compare our results to other recently developed methods.

This paper presents a new version of the upwind compact finite difference scheme for solving the incompressible Navier-Stokes equations in generalized curvilinear coordinates. The artificial compressibility approach is used, which transforms the elliptic-parabolic equations into the hyperbolic-parabolic ones so that flux difference splitting can be applied. The convective terms are approximated by a third-order upwind compact scheme implemented with flux difference splitting, and the viscous terms are approximated by a fourth-order central compact scheme. The solution algorithm used is the Beam-Warming approximate factorization scheme. Numerical solutions to benchmark problems of the steady plane Couette-Poiseuille flow, the lid-driven cavity flow, and the constricting channel flow with varying geometry are presented. The computed results are found in good agreement with established analytical and numerical results. The third-order accuracy of the scheme is verified on uniform rectangular meshes.

Flows containing steady or nearly steady strong shocks on parts of the flow field, and unsteady turbulence with shocklets on other parts of the flow field are difficult to capture accurately and efficiently employing the same numerical scheme, even under the multiblock grid or adaptive grid refinement framework. While sixth-order or higher-order shock-capturing methods are appropriate for unsteady turbulence with shocklets, third-order or lower shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence. In order to minimize the short comings of low order and high order shock-capturing schemes for the subject flows, a multiblock overlapping grid with different types of spatial schemes and orders of accuracy on different blocks is proposed. The recently developed single block high order filter scheme in generalized geometries for Navier Stokes and magnetohydrodynamics systems is extended to multiblock overlapping grid geometries. The first stage in validating the high order overlapping approach with several test cases is included.

We investigate blow-up of the focusing nonlinear Schr ¨odinger equation, in the critical and supercritical cases. Numerical simulations are performed to examine the dependence of the time at which blow-up occurs on properties of the data or the equation. Three cases are considered: dependence on the scale of the nonlinearity when the initial data are fixed; dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed; and dependence upon a damping factor when the initial data are fixed. In most of these situations, monotonicity in the evolution of the blow-up time does not occur.

An implicit non-linear lower-upper symmetric Gauss-Seidel (LU-SGS) solution algorithm has been developed for a high-order spectral difference Navier-Stokes solver on unstructured hexahedral grids. The non-linear LU-SGS solver is preconditioned by a block element matrix, and the system of equations is then solved with the LU decomposition approach. The large sparse Jacobian matrix is computed numerically, resulting in extremely simple operations for arbitrarily complex residual operators. Several inviscid and viscous test cases were performed to evaluate the performance. The implicit solver has shown speedup of 1 to 2 orders of magnitude over the multi-stage Runge-Kutta time integration scheme. 

It is demonstrated that spectral methods can be used to improve the accuracy of numerical solutions obtained by some lower order methods. More precisely, we can use spectral methods to postprocess numerical solutions of initial value differential equations. After a few number of iterations (say 3 to 4), the errors can decrease to a few orders of magnitude less. The iteration uses the Gauss-Seidel type strategy, which gives an explicit way of postprocessing. Numerical examples for ODEs, Hamiltonian system and integral equations are provided. They all indicate that the spectral processing technique can be a very useful way in improving the accuracy of the numerical solutions. In particular, for a Hamiltonian system accuracy is only one of the issues; some other conservative properties are even more important for large time simulations. The spectral postprocessing with the coarse-mesh symplectic initial guess can not only produce high accurate approximations but can also save a significant amount of computational time over the standard symplectic schemes.

We develop the theory for a robust and efficient adaptive multi-element generalized polynomial chaos (ME-gPC) method for elliptic equations with random coefficients for a moderate number (O(10)) of random dimensions. We employ low-order (p ≤ 3) polynomial chaos and refine the solution using adaptivity in the parametric space. We first study the approximation error of ME-gPC and prove its hp-convergence. We subsequently generate local and global a posteriori error estimators. In order to resolve the error equations efficiently, we construct a reduced space using much smaller number of terms in the enhanced polynomial chaos space to capture the errors of ME-gPC approximation. Based on the a posteriori estimators, we propose and implement an adaptive ME-gPC algorithm for elliptic problems with random coefficients. Numerical results for convergence and efficiency are also presented.

In this paper, we consider the application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system. The method in this paper extends the local discontinuous Galerkin method in [10] to the more general application system which is coupled with the Allen-Cahn and Cahn-Hilliard equations. Similar energy stability result as that in [10] is presented. Numerical results for the nonlinear problems which include the Allen-Cahn/Cahn-Hilliard system for one-dimensional and two-dimensional cases demonstrate the accuracy and capability of the numerical method. 

We extend the weighted essentially non-oscillatory (WENO) schemes on two dimensional triangular meshes developed in [7] to three dimensions, and construct a third order finite volume WENO scheme on three dimensional tetrahedral meshes. We use the Lax-Friedrichs monotone flux as building blocks, third order reconstructions made from combinations of linear polynomials which are constructed on diversified small stencils of a tetrahedral mesh, and non-linear weights using smoothness indicators based on the derivatives of these linear polynomials. Numerical examples are given to demonstrate stability and accuracy of the scheme.