Legendre Gauss Lobatto Research Articles

Efficient Spectral Collocation Algorithm for Solving Parabolic Inverse Problems

This paper reports a new Legendre–Gauss–Lobatto collocation (SL-GL-C) method to solve numerically two partial parabolic inverse problems subject to initial-boundary conditions. The problem is reformulated by eliminating the unknown functions using some special assumptions based on Legendre–Gauss—Lobatto quadrature rule. The SL-GL-C is utilized to solve nonclassical parabolic initial-boundary value problems. Accordingly, the inverse problem is reduced into a system of ordinary differential equations (ODEs) and afterwards, such system can be solved numerically using implicit Runge–Kutta (IRK) method of order four. Four examples are introduced to demonstrate the applicability, validity, effectiveness and stable approximations of the present method. Numerical results show the exponential convergence property and error characteristics of presented method.

Spectral element method for band-structure calculations of 3D phononic crystals

The spectral element method (SEM) is a special kind of high-order finite element method (FEM) which combines the flexibility of a finite element method with the accuracy of a spectral method. In contrast to the traditional FEM, the SEM exhibits advantages in the high-order accuracy as the error decreases exponentially with the increase of interpolation degree by employing the Gauss–Lobatto–Legendre (GLL) polynomials as basis functions. In this study, the spectral element method is developed for the first time for the determination of band structures of 3D isotropic/anisotropic phononic crystals (PCs). Based on the Bloch theorem, we present a novel, intuitive discretization formulation for Navier equation in the SEM scheme for periodic media. By virtue of using the orthogonal Legendre polynomials, the generalized eigenvalue problem is converted to a regular one in our SEM implementation to improve the efficiency. Besides, according to the specific geometry structure, 8-node and 27-node hexahedral elements as well as an analytic mesh have been used to accurately capture curved PC models in our SEM scheme. To verify its accuracy and efficiency, this study analyses the phononic-crystal plates with square and triangular lattice arrangements, and the 3D cubic phononic crystals consisting of simple cubic (SC), bulk central cubic (BCC) and faced central cubic (FCC) lattices with isotropic or anisotropic scatters. All the numerical results considered demonstrate that SEM is superior to the conventional FEM and can be an efficient alternative method for accurate determination of band structures of 3D phononic crystals.

Improving the accuracy of mass-lumped finite-elements in the first-order formulation of the wave equation by defect correction

Finite-element discretizations of the acoustic wave equation in the time domain often employ mass lumping to avoid the cost of inverting a large sparse mass matrix. For the second-order formulation of the wave equation, mass lumping on Legendre–Gauss–Lobatto points does not harm the accuracy. Here, we consider a first-order formulation of the wave equation. In that case, the numerical dispersion for odd-degree polynomials exhibits super-convergence with a consistent mass matrix but mass lumping destroys that property. We consider defect correction as a means to restore the accuracy, in which the consistent mass matrix is approximately inverted using the lumped one as preconditioner. For the lowest-degree element on a uniform mesh, fourth-order accuracy in 1D can be obtained with just a single iteration of defect correction.The numerical dispersion curve describes the error in the eigenvalues of the discrete set of equations. However, the error in the eigenvectors also play a role, in two ways. For polynomial degrees above one and when considering a 1-D mesh with constant element size and constant material properties, a number of modes, equal to the maximum polynomial degree, are coupled. One of these is the correct physical mode that should approximate the true eigenfunction of the operator, the other are spurious and should have a small amplitude when the true eigenfunction is projected onto them. We analyze the behaviour of this error as a function of the normalized wavenumber in the form of the leading terms in its series expansion and find that this error exceeds the dispersion error, except for the lowest degree where the eigenvector error is zero. Numerical 1-D tests confirm this behaviour.We briefly analyze the 2-D case, where the lowest-degree polynomial also appears to provide fourth-order accuracy with defect correction, if the grid of squares or triangles is highly regular and material properties are constant.

Computation of Some Second Order Sturm-Liouville Bvps using Chebyshev-Legendre Collocation Method

We propose Chebyshev-Legendre spectral collocation method for solving second order linear and nonlinear eigenvalue problems exploiting Legendre derivative matrix. The Sturm-Liouville (SLP) problems are formulated utilizing Chebyshev-Gauss-Lobatto (CGL) nodes instead of Legendre Gauss-Lobatto (LGL) nodes and Legendre polynomials are taken as basis function. We discuss, in details, the formulations of the present method for the Sturm-Liouville problems (SLP) with Dirichlet and mixed type boundary conditions. The accuracy of this method is demonstrated by computing eigenvalues of three regular and two singular SLP's. Nonlinear Bratu type problem is also tested in this article. The numerical results are in good agreement with the other available relevant studies.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 95-112

GPU-accelerated discontinuous Galerkin methods on hybrid meshes

We present a time-explicit discontinuous Galerkin (DG) solver for the time-domain acoustic wave equation on hybrid meshes containing vertex-mapped hexahedral, wedge, pyramidal and tetrahedral elements. Discretely energy-stable formulations are presented for both Gauss–Legendre and Gauss–Legendre–Lobatto (Spectral Element) nodal bases for the hexahedron. Stable timestep restrictions for hybrid meshes are derived by bounding the spectral radius of the DG operator using order-dependent constants in trace and Markov inequalities. Computational efficiency is achieved under a combination of element-specific kernels (including new quadrature-free operators for the pyramid), multi-rate timestepping, and acceleration using Graphics Processing Units.

WITHDRAWN: Legendre–Gauss–Lobatto collocation method for solving multi-dimensional Fredholm integral equations

Abstract This paper reports a new spectral collocation technique for solving second kind Fredholm integral equations (FIEs). We develop a collocation scheme to approximate FIEs by means of the shifted Legendre–Gauss–Lobatto collocation (SL–GL-C) method. Moreover, we adapt the SL–GL-C algorithm to solve one dimensional second kind FIEs and system of FIEs. Two numerical algorithms are also investigated to solve the two dimensional FIEs. Extensive numerical tests illustrate the capability and high accuracy of the proposed methodologies.

Static analysis of sandwich panels with non-homogeneous soft-cores by novel weak form quadrature element method

The extended high order sandwich panel theory (EHSAPT) is employed to analyze the static behavior of sandwich beams with non-homogeneous soft-cores. The material properties of the orthotropic core vary along the thickness direction according to a power-law form starting from its middle plane to the upper or bottom faces. Due to complexity of the theory, the solutions are obtained numerically by using the weak form quadrature element method. Based on the expanded EHSAPT and differential quadrature rule, a novel N-node weak form quadrature sandwich beam element with Gauss Lobatto Legendre (GLL) nodes is established. The element stiffness matrix and work equivalent load are obtained by using GLL quadrature. Detailed formulations are worked out. Different materials for the face sheets and core of the sandwich beam structure are considered. Numerical results are presented to investigate effects of the power-law exponents, boundary conditions, different types of material system and loading on the displacements and stresses of the sandwich panel with non-homogeneous soft core.

Thermoelastic wave characteristics in a hollow cylinder using the modified wave finite element method

This paper represents a modified formulation of the wave finite element (WFE) method for propagating analysis of thermoelastic waves in a hollow cylinder without energy dissipation. The 2D-high-order spectral element with the Gauss–Legendre–Lobatto integration is applied into the WFE method, which produces the diagonal mass matrix. Based on the assumption of harmonic displacement fields by Fourier series expansion, the general discretization wave equation is simplified from the 3D problem to 2D. Dispersion properties of elastic wave propagation in the hollow cylinder are computed considering the choice of the spectral element orders, and the results indicate the high efficiency and high accuracy of the modified formulation compared with that of the software Disperse. Then, using the modified formulation, the thermoelastic dynamic equation of the cylinder is derived from the generalized thermoelasticity theory. The propagation of the thermoelastic wave (including two kinds of wave modes) in the cylinder without energy dissipation is discussed in different cases. Finally, wave structures along the radial direction of thermoelastic wave modes are shown at the nondimensional frequency 1.25, which can be used for the recognition of different modes.

Note on Using Radial Basis Functions Method for Solving Nonlinear Integral Equations

In this note, the solution of nonlinear integral equations was discussed using radial basis functions (RBFs) method. This method will represent the solution of nonlinear integral equation by interpolating the RBFs based on Legendre-Gauss-Lobatto (LGL) nodes and weights. Zeros of the shifted Legendre polynomials are used as the collocation points. The method is used to some examples to illustrate the accuracy and the implementation of the method.

Entropy Stable Staggered Grid Discontinuous Spectral Collocation Methods of any Order for the Compressible Navier--Stokes Equations

Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for the compressible Euler and Navier--Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [M. H. Carpenter, T. C. Fisher, E. J. Nielsen, and S. H. Frankel, SIAM J. Sci. Comput., 36 (2014), pp. B835--B867, M. Parsani, M. H. Carpenter, and E. J. Nielsen, J. Comput. Phys., 292 (2015), pp. 88--113], extends the applicable set of points from tensor product, Legendre--Gauss--Lobatto (LGL), to a combination of tensor product Legendre--Gauss (LG) and LGL points. The new semidiscrete operators discretely conserve mass, momentum, energy, and satisfy a mathematical entropy inequality for the compressible Navier--Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG ope...

Symplectic Irregular Interpolation Algorithms for Optimal Control Problems

Symplectic numerical methods for optimal control problems with irregular interpolation schemes are developed and the comparisons between irregular interpolation schemes and equidistant scheme are made in this paper. The irregular interpolation points, which are the collocation points usually adopted by pseudospectral (PS) methods, such as Legendre–Gauss, Legendre–Gauss–Radau, Legendre–Gauss–Lobatto and Chebyshev–Gauss–Lobatto points, are taken into consideration in this study. The symplectic numerical method with irregular points is proposed firstly. Then, several examples with different complexities highlight the differences in performance between different kinds of interpolation schemes. The numerical results show that the convergence of the present symplectic numerical methods can be obtained by increasing the number of sub-intervals or the number of interpolation points. Moreover, the comparison results show that the convergence of the symplectic numerical methods are generally independent on the type of interpolation points and the computational efficiency is not sensitive to the choice of interpolation points in general. Thus, the symplectic numerical methods with different interpolation schemes have obvious difference with the PS methods.

Efficient and accurate computation of model predictive control using pseudospectral discretization

The model predictive control (MPC) is implemented by repeatedly solving an open loop optimal control problem (OCP). For the real-time implementation, the OCP is often discretized with evenly spaced time grids. This evenly spaced discretization, however, is accurate only if sufficiently small sampling time is used, which leads to heavy computational load. This paper presents a method to efficiently and accurately compute the continuous-time MPC problem based on the pseudospectral discretization, which utilizes unevenly spaced collocation points. The predictive horizon is virtually doubled by augmenting a mirrored horizon such that denser collocation points can be used towards the current time step, and sparser points can be used towards the end time of predictive horizon. Then, both state and control variables are approximated by Lagrange polynomials at only a half of LGL (Legendre–Gauss–Lobatto) collocation points. This implies that high accuracy can be achieved with a much less number of collocation points, which results in much reduced computational load. Examples are used to demonstrate its advantages over the evenly spaced discretization.

Solution of nonlinear weakly singular Volterra integral equations using the fractional‐order Legendre functions and pseudospectral method

In this article, our main goal is to render an idea to convert a nonlinear weakly singular Volterra integral equation to a non‐singular one by new fractional‐order Legendre functions. The fractional‐order Legendre functions are generated by change of variable on well‐known shifted Legendre polynomials. We consider a general form of singular Volterra integral equation of the second kind. Then the fractional Legendre–Gauss–Lobatto quadratures formula eliminates the singularity of the kernel of the integral equation. Finally, the Legendre pseudospectral method reduces the solution of this problem to the solution of a system of algebraic equations. This method also can be utilized on fractional differential equations as well. The comparison of results of the presented method and other numerical solutions shows the efficiency and accuracy of this method. Also, the obtained maximum error between the results and exact solutions shows that using the present method leads to accurate results and fast convergence for solving nonlinear weakly singular Volterra integral equations. Copyright © 2015 John Wiley & Sons, Ltd.

Using spectral element method to solve variational inequalities with applications in finance

Under the Black–Scholes model, the value of an American option solves a time dependent variational inequality problem (VIP). In this paper, first we discretize the variational inequality of American option in temporal direction by applying the Rannacher time stepping and achieve a sequence of elliptic variational inequalities. Second we discretize the spatial domain of variational inequalities by using spectral element methods with high order Lagrangian polynomials introduced on Gauss–Legendre–Lobatto points. Also by computing integrals by the Gauss–Legendre–Lobatto quadrature rule we derive a sequence of the linear complementarity problems (LCPs) having a positive definite sparse coefficient matrix. To find the unique solutions of the LCPs, we use the projected successive over-relaxation (PSOR) algorithm. Furthermore we present some existence and uniqueness theorems for the variational inequalities and LCPs. Finally, theoretical results are verified on the relevant numerical examples.

The Mixed Spectral-Element Method for Anisotropic, Lossy, and Open Waveguides

The higher order mixed spectral-element method (SEM) is proposed for the analysis of anisotropic, lossy, and open waveguides. The new variational formulation incorporating Gauss’ law into the vectorial wave equation is completely free of spurious modes. It utilizes the edge-based curl-conforming Gauss–Lobatto–Legendre (GLL) polynomials to approximate the tangential vector of the electric field and the nodal-based scalar GLL basis functions to discretize its longitudinal component to obtain the highly accurate solution. By employing the mass-lumping technique, only a smaller generalized eigenvalue problem needs to be solved. Several numerical examples are given to verify that the mixed SEM is free of any spurious eigenmode and has spectral accuracy with the propagation constant.

Efficient Spectral Collocation Algorithm for a Two-Sided Space Fractional Boussinesq Equation with Non-local Conditions

A spectral shifted Legendre Gauss–Lobatto collocation method is developed and analyzed to solve numerically one-dimensional two-sided space fractional Boussinesq (SFB) equation with non-classical boundary conditions. The method depends basically on the fact that an expansion in a series of shifted Legendre polynomials $${P_{L,n}(x), \ x\in[0,L]}$$ is assumed, for the function and its space-fractional derivatives occurring in the two-sided SFB equation. The Legendre–Gauss–Lobatto quadrature rule is established to treat the non-local conservation conditions, and then the problem with its non-local conservation conditions is reduced to a system of ordinary differential equations (ODEs) in time. Thereby, the expansion coefficients are then determined by reducing the two-sided SFB with its boundary and initial conditions to a system of ODEs for these coefficients. This system may be solved numerically in a step-by-step manner by using implicit Runge–Kutta method of order four. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented.

A highly accurate collocation algorithm for 1 + 1 and 2 + 1 fractional percolation equations

This paper reports two successive spectral collocation methods, that enable easy and highly accuracy discretization, for 1 + 1 and 2 + 1 fractional percolation equations (FPEs). The first step depends mainly on the shifted Legendre Gauss–Lobatto collocation method for spatial discretization. An expansion in a series of shifted Legendre polynomials for the approximate solution and its spatial derivatives occurring in the FPE is investigated. In addition, the Legendre-Gauss–Lobatto quadrature rule is established to treat the boundary conditions. Thereby, the expansion coefficients are then determined by reducing the FPE with its boundary conditions to a system of ordinary differential equations for these coefficients. The second step is to propose the shifted Chebyshev Gauss–Radau collocation scheme, for temporal discretization, to reduce such a system to a system of algebraic equations, which is far easier to solve. The proposed collocation scheme, both in temporal and spatial discretizations, is successfully extended to the numerical solution of two-dimensional FPEs. An upper bound of the absolute error is obtained of the approximate solution for the two-dimensional case. Convergence properties of the method are discussed through numerical examples. Several numerical examples with comparisons are reported to highlight the high accuracy of the present method over other numerical techniques.

A Unified Pseudospectral Computational Framework for Optimal Control of Road Vehicles

This paper presents a unified pseudospectral computational framework for accurately and efficiently solving optimal control problems (OCPs) of road vehicles. Under this framework, any continuous-time OCP is converted into a nonlinear programming (NLP) problem via pseudospectral transformation, in which both states and controls are approximated by global Lagrange interpolating polynomials at Legendre–Gauss–Lobatto (LGL) collocation points. The mapping relationship between the costates of OCP and the KKT multipliers of NLP is derived for checking the optimality of solutions. For the sake of engineering practice, a quasi-Newton iterative algorithm is integrated to accurately calculate the LGL points, and a multiphase preprocessing strategy is proposed to handle nonsmooth problems. A general solver called pseudospectral OCP solver (POPS) is developed in MATLAB environment to implement the computational framework. Finally, two classic vehicle automation problems are formulated and numerically solved by POPS: 1) optimization of ecodriving strategy in hilly road conditions; and 2) optimal path planning in an overtaking scenario. The comparison with an equally spaced direct method is presented to show the effectiveness of this unified framework.

A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations

A shifted Legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations (TFSEs) subject to initial-boundary and non-local conditions. The first step depends mainly on shifted Legendre Gauss–Lobatto collocation (SL-GL-C) method for spatial discretization; an expansion in a series of shifted Legendre polynomials for the approximate solution and its spatial derivatives occurring in the TFSE is investigated. In addition, the Legendre–Gauss–Lobatto quadrature rule is established to treat the nonlocal conservation conditions. Thereby, the expansion coefficients are then determined by reducing the TFSE with its nonlocal conditions to a system of fractional differential equations (SFDEs) for these coefficients. The second step is to propose a shifted Legendre Gauss–Radau collocation (SL-GR-C) scheme, for temporal discretization, to reduce such system into a system of algebraic equations which is far easier to be solved. The proposed collocation scheme, both in temporal and spatial discretizations, is successfully extended to solve the two-dimensional TFSE. Numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms. By selecting relatively limited Legendre Gauss–Lobatto and Gauss–Radau collocation nodes, we are able to get very accurate approximations, demonstrating the utility and high accuracy of the new approach over other numerical methods.

A Legendre Polynomials Based Method for Stability Analysis of Milling Processes

This paper presents a time-domain approach for a semi-analytical prediction of stability in milling using the Legendre polynomials. The governing equation of motion of milling processes is expressed as a delay-differential equation (DDE) with time periodic coefficients. After the DDE being re-expressed in state-space form, the state vector is approximated by a series of Legendre polynomials. With the help of the Legendre–Gauss–Lobatto (LGL) quadrature, a discrete dynamic map is formulated to approximate the original DDE, and utilized to predict the milling stability based on Floquet theory. With numerical examples illustrating the efficiency and accuracy of the proposed approach, an experimental example validates the method.