Showing papers in "Journal of Scientific Computing in 2014"

A Splitting Method for Orthogonality Constrained Problems

Abstract: Orthogonality constrained problems are widely used in science and engineering. However, it is challenging to solve these problems efficiently due to the non-convex constraints. In this paper, a splitting method based on Bregman iteration is represented to tackle the optimization problems with orthogonality constraints. With the proposed method, the constrained problems can be iteratively solved by computing the corresponding unconstrained problems and orthogonality constrained quadratic problems with analytic solutions. As applications, we demonstrate the robustness of our method in several problems including direction fields correction, noisy color image restoration and global conformal mapping for genus-0 surfaces construction. Numerical comparisons with existing methods are also conducted to illustrate the efficiency of our algorithms.

A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem

Abstract: We present a novel finite element method for the Stokes problem on fictitious domains. We prove inf-sup stability, optimal order convergence and uniform boundedness of the condition number of the discrete system. The finite element formulation is based on a stabilized Nitsche method with ghost penalties for the velocity and pressure to obtain stability in the presence of small cut elements. We demonstrate for the first time the applicability of the Nitsche fictitious domain method to three-dimensional Stokes problems. We further discuss a general, flexible and freely available implementation of the method and present numerical examples supporting the theoretical results.

A New Error Analysis of Crank---Nicolson Galerkin FEMs for a Generalized Nonlinear Schrödinger Equation

Abstract: In this paper, we study linearized Crank---Nicolson Galerkin FEMs for a generalized nonlinear Schrodinger equation. We present the optimal $$L^2$$ L 2 error estimate without any time-step restrictions, while previous works always require certain conditions on time stepsize. A key to our analysis is an error splitting, in terms of the corresponding time-discrete system, with which the error is split into two parts, the temporal error and the spatial error. Since the spatial error is $$\tau $$ ? -independent, the numerical solution can be bounded in $$L^{\infty }$$ L ? -norm by an inverse inequality unconditionally. Then, the optimal $$L^2$$ L 2 error estimate can be obtained by a routine method. To confirm our theoretical analysis, numerical results in both two and three dimensional spaces are presented.

Lower Bounds for Eigenvalues of Elliptic Operators: By Nonconforming Finite Element Methods

Abstract: The paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The conclusion is that if local approximation properties of nonconforming finite element spaces are better than total errors (sums of global approximation errors and consistency errors) of nonconforming finite element methods, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we analyze abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. We show that this condition hold for most low-order nonconforming finite elements in literature. In addition, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we also propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally, we prove the saturation condition for most nonconforming elements.

A Linear Iteration Algorithm for a Second-Order Energy Stable Scheme for a Thin Film Model Without Slope Selection

Abstract: We present a linear iteration algorithm to implement a second-order energy stable numerical scheme for a model of epitaxial thin film growth without slope selection. The PDE, which is a nonlinear, fourth-order parabolic equation, is the $$L^2$$ L 2 gradient flow of the energy $$ \int _\Omega \left( - \frac{1}{2} \ln \left( 1 + | abla \phi |^2 \right) + \frac{\epsilon ^2}{2}|\Delta \phi (\mathbf{x})|^2 \right) \mathrm{d}\mathbf{x}$$ ? Ω - 1 2 ln 1 + | ? ? | 2 + ∈ 2 2 | Δ ? ( x ) | 2 d x . The energy stability is preserved by a careful choice of the second-order temporal approximation for the nonlinear term, as reported in recent work (Shen et al. in SIAM J Numer Anal 50:105---125, 2012). The resulting scheme is highly nonlinear, and its implementation is non-trivial. In this paper, we propose a linear iteration algorithm to solve the resulting nonlinear system. To accomplish this we introduce an $$O(s^2)$$ O ( s 2 ) (with $$s$$ s the time step size) artificial diffusion term, a Douglas-Dupont-type regularization, that leads to a contraction mapping property. As a result, the highly nonlinear system can be decomposed as an iteration of purely linear solvers, which can be very efficiently implemented with the help of FFT in a collocation Fourier spectral setting. We present a careful analysis showing convergence for the numerical scheme in a discrete $$L^\infty (0, T; H^1) \cap L^2 (0,T; H^3)$$ L ? ( 0 , T ; H 1 ) ? L 2 ( 0 , T ; H 3 ) norm. Some numerical simulation results are presented to demonstrate the efficiency of the linear iteration solver and the convergence of the scheme as a whole.

High-Order Multiderivative Time Integrators for Hyperbolic Conservation Laws

Abstract: Multiderivative time integrators have a long history of development for ordinary differential equations, and yet to date, only a small subset of these methods have been explored as a tool for solving partial differential equations (PDEs). This large class of time integrators include all popular (multistage) Runge---Kutta as well as single-step (multiderivative) Taylor methods. (The latter are commonly referred to as Lax---Wendroff methods when applied to PDEs). In this work, we offer explicit multistage multiderivative time integrators for hyperbolic conservation laws. Like Lax---Wendroff methods, multiderivative integrators permit the evaluation of higher derivatives of the unknown in order to decrease the memory footprint and communication overhead. Like traditional Runge---Kutta methods, multiderivative integrators admit the addition of extra stages, which introduce extra degrees of freedom that can be used to increase the order of accuracy or modify the region of absolute stability. We describe a general framework for how these methods can be applied to two separate spatial discretizations: the discontinuous Galerkin (DG) method and the finite difference essentially non-oscillatory (FD-WENO) method. The two proposed implementations are substantially different: for DG we leverage techniques that are closely related to generalized Riemann solvers; for FD-WENO we construct higher spatial derivatives with central differences. Among multiderivative time integrators, we argue that multistage two-derivative methods have the greatest potential for multidimensional applications, because they only require the flux function and its Jacobian, which is readily available. Numerical results indicate that multiderivative methods are indeed competitive with popular strong stability preserving time integrators.

Comparison Between Reduced Basis and Stochastic Collocation Methods for Elliptic Problems

Abstract: The stochastic collocation method (Babuska et al. in SIAM J Numer Anal 45(3):1005---1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411---2442, 2008a; SIAM J Numer Anal 46(5):2309---2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118---1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289---294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu , 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229---275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41---44):3187---3206, 2009; Arch Comput Methods Eng 17:435---454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions $$O(1)$$ O ( 1 ) to moderate dimensions $$O(10)$$ O ( 10 ) and to high dimensions $$O(100)$$ O ( 100 ) . The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.

A Comparison Between the Interpolated Bounce-Back Scheme and the Immersed Boundary Method to Treat Solid Boundary Conditions for Laminar Flows in the Lattice Boltzmann Framework

Abstract: In this paper, the interpolated bounce-back scheme and the immersed boundary method are compared in order to handle solid boundary conditions in the lattice Boltzmann method. These two approaches are numerically investigated in two test cases: a rigid fixed cylinder invested by an incoming viscous fluid and an oscillating cylinder in a calm viscous fluid. Findings in terms of velocity profiles in several cross sections are shown. Differences and similarities between the two methods are discussed, by emphasizing pros and cons in terms of stability and computational effort of the numerical algorithm.

Entropy-Stable Schemes for the Euler Equations with Far-Field and Wall Boundary Conditions

Abstract: In this paper entropy-stable numerical schemes for the Euler equations in one space dimension subject to far-field and wall boundary conditions are derived. Furthermore, a stable numerical treatment of interfaces between different grid domains is proposed. Numerical computations with second- and fourth-order accurate schemes corroborate the stability and accuracy of the proposed boundary treatment.

Energy Stable Flux Reconstruction Schemes for Advection---Diffusion Problems on Tetrahedra

Abstract: The flux reconstruction (FR) methodology provides a unifying description of many high-order schemes, including a particular discontinuous Galerkin (DG) scheme and several spectral difference (SD) schemes. In addition, the FR methodology has been used to generate new classes of high-order schemes, including the recently discovered `energy stable' FR schemes. These schemes, which are often referred to as VCJH (Vincent---Castonguay---Jameson---Huynh) schemes, are provably stable for linear advection---diffusion problems in 1D and on triangular elements. The VCJH schemes have been successfully applied to a wide variety of problems in 1D and 2D, ranging from linear advection---diffusion problems, to fluid mechanics problems requiring the solution of the compressible Navier---Stokes equations. Based on the results of these numerical experiments, it has been shown that certain VCJH schemes maintain the expected order of spatial accuracy and possess explicit time-step limits which rival those of the collocation-based nodal DG scheme. However, it remained to be seen whether the VCJH schemes could be extended to 3D on tetrahedral elements, enabling their convenient application to the complex geometries that arise in many real-world problems. For the first time, this article presents an extension of the VCJH schemes to tetrahedral elements. This work provides a formal proof of the stability of the new schemes and assesses their performance via numerical experiments on model problems.

Partitioned and Implicit---Explicit General Linear Methods for Ordinary Differential Equations

Abstract: Implicit---explicit (IMEX) time stepping methods can efficiently solve differential equations with both stiff and nonstiff components. IMEX Runge---Kutta methods and IMEX linear multistep methods have been studied in the literature. In this paper we study new implicit---explicit methods of general linear type. We develop an order conditions theory for high stage order partitioned general linear methods (GLMs) that share the same abscissae, and show that no additional coupling order conditions are needed. Consequently, GLMs offer an excellent framework for the construction of multi-method integration algorithms. Next, we propose a family of IMEX schemes based on diagonally-implicit multi-stage integration methods and construct practical schemes of order up to three. Numerical results confirm the theoretical findings.

A Multigrid Method for Helmholtz Transmission Eigenvalue Problems

Abstract: In this paper, we analyze the convergence of a finite element method for the computation of transmission eigenvalues and corresponding eigenfunctions. Based on the obtained error estimate results, we propose a multigrid method to solve the Helmholtz transmission eigenvalue problem. This new method needs only linear computational work. Numerical results are provided to validate the efficiency of the proposed method.

A Posteriori Error Estimates for Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

Abstract: A residual type a posteriori error estimator is presented and analyzed for Weak Galerkin finite element methods for second order elliptic problems. The error estimator is proved to be efficient and reliable through two estimates, one from below and the other from above, in terms of an $$H^1$$ -equivalent norm for the exact error. Two numerical experiments are conducted to demonstrate the effectiveness of adaptive mesh refinement guided by this estimator.

Acoustic Wave Propagation in Complicated Geometries and Heterogeneous Media

Abstract: We construct finite difference discretizations of the acoustic wave equation in complicated geometries and heterogeneous media. Particular emphasis is placed on the accurate treatment of interfaces at which the underlying media parameters have jump discontinuities. Discontinuous media is treated by subdividing the domain into blocks with continuous media. The equation on each block is then discretized with finite difference operators satisfying a summation-by-parts property and patched together via the simultaneous approximation term method. The energy method is used to estimate a semi-norm of the numerical solution in terms of data, showing that the discretization is stable. Numerical experiments in two and three spatial dimensions verifies the accuracy and stability properties of the schemes.

A C^0-Weak Galerkin Finite Element Method for the Biharmonic Equation

Abstract: A \(C^0\)-weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for \(C^0\) functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete \(H^2\) norm and the standard \(H^1\) and \(L^2\) norms with appropriate regularity assumptions. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimates. This refined interpolation preserves the volume mass of order \((k+1-d)\) and the surface mass of order \((k+2-d)\) for the \(P_{k+2}\) finite element functions in \(d\)-dimensional space.

On the Galerkin/Finite-Element Method for the Serre Equations

Abstract: A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime. The fully-discrete scheme utilizes the Galerkin / finite-element method based on smooth periodic splines in space, and an explicit fourth-order Runge---Kutta method in time. Computations compared with exact solitary and cnoidal wave solutions show that the scheme achieves the optimal orders of accuracy in space and time. These computations also show that the stability of this scheme does not impose very restrictive conditions on the temporal stepsize. In addition, solitary, cnoidal, and dispersive shock waves are studied in detail using this numerical scheme for the Serre system and compared with the `classical' Boussinesq system for small-amplitude shallow water waves. The results show that the interaction of solitary waves in the Serre system is more inelastic. The efficacy of the numerical scheme for modeling dispersive shocks is shown by comparison with asymptotic results. These results have application to the modeling of shallow water waves of intermediate or large amplitude.

High-Order Flux Correction for Viscous Flows on Arbitrary Unstructured Grids

Abstract: A novel high-order method, termed flux correction, previously formulated for inviscid flows, is extended to viscous flows on arbitrary triangular grids. The correction method involves the addition of truncation error-canceling terms to the second-order linear Galerkin (node-centered finite volume) scheme to produce a third-order inviscid and fourth-order viscous scheme. The correction requires minimal modification of the underlying second-order scheme. As such, the method retains many of the advantages of traditional finite volume schemes, including robust shock capturing, low algorithmic complexity, and solver efficiency. In addition, we extend the scheme to unsteady flows. Verification and validation studies in two dimensions are presented. Significant improvement in accuracy is observed in all cases, with between 30---70 % increase in computational cost over a second-order finite volume method.

Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

Abstract: We propose a simple modification of standard weighted essentially non-oscillatory (WENO) finite volume methods for Cartesian grids, which retains the full spatial order of accuracy of the one-dimensional discretization when applied to nonlinear multidimensional systems of conservation laws. We derive formulas, which allow us to compute high-order accurate point values of the conserved quantities at grid cell interfaces. Using those point values, we can compute a high-order flux at the center of a grid cell interface. Finally, we use those point values to compute high-order accurate averaged fluxes at cell interfaces as needed by a finite volume method. The method is described in detail for the two-dimensional Euler equations of gas dynamics. An extension to the three-dimensional case as well as to other nonlinear systems of conservation laws in divergence form is straightforward. Furthermore, similar ideas can be used to improve the accuracy of WENO type methods for hyperbolic systems which are not in divergence form. Several test computations confirm the high-order accuracy for smooth nonlinear problems.

Optimal Strong-Stability-Preserving Runge---Kutta Time Discretizations for Discontinuous Galerkin Methods

Abstract: Discontinuous Galerkin (DG) spatial discretizations are often used in a method-of-lines approach with explicit strong-stability-preserving (SSP) Runge---Kutta (RK) time steppers for the numerical solution of hyperbolic conservation laws. The time steps that are employed in this type of approach must satisfy Courant---Friedrichs---Lewy stability constraints that are dependent on both the region of absolute stability and the SSP coefficient of the RK method. While existing SSPRK methods have been optimized with respect to the latter, it is in fact the former that gives rise to stricter constraints on the time step in the case of RKDG stability. Therefore, in this work, we present the development of new "DG-optimized" SSPRK methods with stability regions that have been specifically designed to maximize the stable time step size for RKDG methods of a given order in one space dimension. These new methods represent the best available RKDG methods in terms of computational efficiency, with significant improvements over methods using existing SSPRK time steppers that have been optimized with respect to SSP coefficients. Second-, third-, and fourth-order methods with up to eight stages are presented, and their stability properties are verified through application to numerical test cases.

Parametrized Maximum Principle Preserving Flux Limiters for High Order Schemes Solving Multi-Dimensional Scalar Hyperbolic Conservation Laws

Abstract: In this paper, we will extend the strict maximum principle preserving flux limiting technique developed for one dimensional scalar hyperbolic conservation laws to the two-dimensional scalar problems. The parametrized flux limiters and their determination from decoupling maximum principle preserving constraint is presented in a compact way for two-dimensional problems. With the compact fashion that the decoupling is carried out, the technique can be easily applied to high order finite difference and finite volume schemes for multi-dimensional scalar hyperbolic problems. For the two-dimensional problem, the successively defined flux limiters are developed for the multi-stage total-variation-diminishing Runge---Kutta time-discretization to improve the efficiency of computation. The high order schemes with successive flux limiters provide high order approximation and maintain strict maximum principle with mild Courant-Friedrichs-Lewy constraint. Two dimensional numerical evidence is given to demonstrate the capability of the proposed approach.

Optimal Error Analysis of Galerkin FEMs for Nonlinear Joule Heating Equations

Abstract: We study in this paper two linearized backward Euler schemes with Galerkin finite element approximations for the time-dependent nonlinear Joule heating equations. By introducing a time-discrete (elliptic) system as proposed in Li and Sun (Int J Numer Anal Model 10:622---633, 2013; SIAM J Numer Anal (to appear)), we split the error function as the temporal error function plus the spatial error function, and then we present unconditionally optimal error estimates of $$r$$ r th order Galerkin FEMs ( $$1 \le r \le 3$$ 1 ≤ r ≤ 3 ). Numerical results in two and three dimensional spaces are provided to confirm our theoretical analysis and show the unconditional stability (convergence) of the schemes.

A Simple Conforming Mixed Finite Element for Linear Elasticity on Rectangular Grids in Any Space Dimension

Abstract: We construct a family of lower-order rectangular conforming mixed finite elements, in any space dimension. In the method, the normal stress is approximated by quadratic polynomials $$\{1, x_{i}, x_{i}^{2}\}$$ { 1 , x i , x i 2 } , the shear stress by bilinear polynomials $$\{1, x_{i}, x_{j}, x_{i}x_{j}\}$$ { 1 , x i , x j , x i x j } , and the displacement by linear polynomials $$\{1, x_{i} \}$$ { 1 , x i } . The number of total degrees of freedom (dof) per element is 10 plus 4 in 2D, and 21 plus 6 in 3D, while the previous record of least dof for conforming element is 17 plus 4 in 2D, and 72 plus 12 in 3D. The feature of this family of elements is, besides simplicity, that shape function spaces for both stress and displacement are independent of the spatial dimension $$n$$ n . As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. The well-posedness condition and the optimal a priori error estimate are proved. Numerical tests show the stability and effectiveness of these new elements.

Folding-Free Global Conformal Mapping for Genus-0 Surfaces by Harmonic Energy Minimization

Abstract: Surface conformal maps between genus-0 surfaces play important roles in applied mathematics and engineering, with applications in medical image analysis and computer graphics. Previous work (Gu and Yau in Commun Inf Syst 2(2):121---146, 2002) introduces a variational approach, where global conformal parameterization of genus-0 surfaces was addressed through minimizing the harmonic energy, with two weaknesses: its gradient descent iteration is slow, and its solutions contain undesired parameterization foldings when the underlying surface has long sharp features. In this paper, we propose an algorithm that significantly accelerates the harmonic energy minimization and a method that iteratively removes foldings by taking advantages of the weighted Laplace---Beltrami eigen-projection. Experimental results show that the proposed approaches compute genus-0 surface harmonic maps much faster than the existing algorithm in Gu and Yau (Commun Inf Syst 2(2):121---146, 2002) and the new results contain no foldings.

Optimal Point-Wise Error Estimate of a Compact Difference Scheme for the Coupled Gross---Pitaevskii Equations in One Dimension

Abstract: The coupled Gross---Pitaevskii (CGP) equation studied in this paper is an important mathematical model describing two-component Bose---Einstein condensate with an internal atomic Josephson junction. We here analyze a compact finite difference scheme which conserves the total mass and energy in the discrete level for the CGP equation. In general, due to the difficulty caused by compact difference on nonlinear terms, optimal point-wise error estimates without any restrictions on the grid ratios of compact difference schemes for nonlinear partial differential equations are very hard to be established. To overcome the difficulty caused by the compact difference operator, we introduce a new norm and an interesting transformation by which the difference scheme is transformed into a special equivalent vector form, we then use the energy method and some important lemmas on the equivalent system to obtain the optimal convergent rate, without any restrictions on the grid ratio, at the order of $$O(h^{4}+\tau ^2)$$ O ( h 4 + ? 2 ) in the maximum norm with time step $$\tau $$ ? and mesh size $$h$$ h . Finally, numerical results are reported to test the theoretical results and simulate the dynamics of the CGP equation.

A Symmetric and Consistent Immersed Finite Element Method for Interface Problems

Abstract: The non-conforming immersed finite element method (IFEM) developed in Li et al. (Numer Math 96:61---98, 2003) for interface problems is extensively studied in this paper. The non-conforming IFEM is very much like the standard finite element method but with modified basis functions that enforce the natural jump conditions on interface elements. While the non-conforming IFEM is simple and has reasonable accuracy, it is not fully second order accurate due to the discontinuities of the modified basis functions. While the conforming IFEM also developed in Li et al. (Numer Math 96:61---98, 2003) is fully second order accurate, the implementation is more complicated. A new symmetric and consistent IFEM has been developed in this paper. The new method maintains the advantages of the non-conforming IFEM by using the same basis functions but it is symmetric, consistent, and more important, it is second order accurate. The idea is to add some correction terms to the weak form to take into account of the discontinuities in the basis functions. Optimal error estimates are derived for the new symmetric and consistent IFE method in the $$L^2$$ L 2 and $$H^1$$ H 1 norms. Numerical examples presented in this paper confirm the theoretical analysis and show that the new developed IFE method has $$O(h^2)$$ O ( h 2 ) convergence in the $$L^\infty $$ L ? norm as well.

Variational Space---Time Methods for the Wave Equation

Abstract: In this work we present some new variational space---time discretisations for the scalar-valued acoustic wave equation as a prototype model for the vector-valued elastic wave equation. The second-order hyperbolic equation is rewritten as a first-order in time system of equations for the displacement and velocity field. For the discretisation in time we apply continuous Galerkin---Petrov and discontinuous Galerkin methods, and for the discretisation in space we apply the symmetric interior penalty discontinuous Galerkin method. The resulting algebraic system of equations exhibits a block structure. First, it is simplified by some calculations to a linear system for one of the variables and a vector update for the other variable. Using the block diagonal structure of the mass matrix from the discontinuous Galerkin discretisation in space, the reduced system can be condensed further such that the overall linear system can be solved efficiently. The convergence behaviour of the presented schemes is studied carefully by numerical experiments. Moreover, the performance and stability properties of the schemes are illustrated by a more sophisticated problem with complex wave propagation phenomena in heterogeneous media.

The Relationships Between Chebyshev, Legendre and Jacobi Polynomials: The Generic Superiority of Chebyshev Polynomials and Three Important Exceptions

Abstract: We analyze the asymptotic rates of convergence of Chebyshev, Legendre and Jacobi polynomials. One complication is that there are many reasonable measures of optimality as enumerated here. Another is that there are at least three exceptions to the general principle that Chebyshev polynomials give the fastest rate of convergence from the larger family of Jacobi polynomials. When $$f(x)$$ f ( x ) is singular at one or both endpoints, all Gegenbauer polynomials (including Legendre and Chebyshev) converge equally fast at the endpoints, but Gegenbauer polynomials converge more rapidly on the interior with increasing order $$m$$ m . For functions on the surface of the sphere, associated Legendre functions, which are proportional to Gegenbauer polynomials, are best for the latitudinal dependence. Similarly, for functions on the unit disk, Zernike polynomials, which are Jacobi polynomials in radius, are superior in rate-of-convergence to a Chebyshev---Fourier series. It is true, as was conjectured by Lanczos 60 years ago, that excluding these exceptions, the Chebyshev coefficients $$a_{n}$$ a n usually decrease faster than the Legendre coefficients $$b_{n}$$ b n by a factor of $$\sqrt{n}$$ n . We calculate the proportionality constant for a few examples and restrictive classes of functions. The more precise claim that $$b_{n} \sim \sqrt{\pi /2} \sqrt{n} a_{n}$$ b n ~ ? / 2 n a n , made by Lanczos and later Fox and Parker, is true only for rather special functions. However, individual terms in the large $$n$$ n asymptotics of Chebyshev and Legendre coefficients usually do display this proportionality.

Approximation of the Stokes---Darcy System by Optimization

Abstract: A solution algorithm for the linear/nonlinear Stokes---Darcy coupled problem is proposed and investigated. The coupled system is formulated as a constrained optimal control problem, where a flow balance is forced across the interface, inflow, and outflow boundaries by minimizing a suitably defined functional. Optimization is achieved by exploiting a Neumann type boundary condition imposed on each subproblem as a control. A numerical algorithm is presented for a least squares functional whose solution yields a minimizer of the constrained optimization problem. Numerical experiments are provided to validate accuracy and efficiency of the algorithm.

An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Triangular Elements

Abstract: The flux reconstruction approach offers an efficient route to high-order accuracy on unstructured grids. The location of the solution points plays an important role in determining the stability and accuracy of FR schemes on triangular elements. In particular, it is desirable that a solution point set (i) defines a well conditioned nodal basis for representing the solution, (ii) is symmetric, (iii) has a triangular number of points and, (iv) minimises aliasing errors when constructing a polynomial representation of the flux. In this paper we propose a methodology for generating solution points for triangular elements. Using this methodology several thousand point sets are generated and analysed. Numerical performance is assessed through an Euler vortex test case. It is found that the Lebesgue constant and quadrature strength of the points are strong indicators of stability and performance. Further, at polynomial orders $$\wp = 4,6,7$$ ? = 4 , 6 , 7 solution points with superior performance to those tabulated in literature are discovered.

Long-Term Stability of Multi-Value Methods for Ordinary Differential Equations

Abstract: Much effort is put into the construction of general linear methods with the aim of achieving an excellent long-time behavior for the integration of Hamiltonian systems. In this article, a backward error analysis is presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. For carefully constructed methods (symmetric and zero growth parameters) the error in the parasitic components typically grows like $$h^{p+4}\exp (h^2Lt)$$ h p + 4 exp ( h 2 L t ) , where $$p$$ p is the order of the method, and $$L$$ L depends on the problem and on the coefficients of the method. This is confirmed by numerical experiments.