# A fully spectral method for hyperbolic equations

15 Jan 1996-International Journal for Numerical Methods in Engineering (Wiley)-Vol. 39, Iss: 1, pp 67-79

About: This article is published in International Journal for Numerical Methods in Engineering.The article was published on 1996-01-15. It has received 4 citations till now. The article focuses on the topics: Upwind scheme & Hyperbolic partial differential equation.

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TL;DR: A parallel spectral algorithm is developed for hyperbolic initial boundary value problems in one space dimension and it is shown that for the case of analytic coefficients and data, satisfying all the required compatibility conditions so that the solution is analytic, the numerical solution is exponentially accurate in N.

Abstract: In this paper a parallel spectral algorithm is developed for hyperbolic initial boundary value problems in one space dimension. The Galerkin-Collocation method, which is spectrally accurate in both space and time, is parallelized by using domain decomposition. This procedure leads to a minimization problem in which there is coupling at inter-domain boundaries. We construct a decoupled preconditioner which can be used to iteratively solve the minimization problem. Symmetric formulation of the problem, which is needed to compute the residual for the normal equations, is discussed. The methodology outlined for computing the normal equations applies equally well to computation of the residual for the p and h–p versions of the finite element method. There is, therefore, no need to compute the mass and stiffness matrices to obtain the residual, as is normally done. This leads to a great saving in time and memory particularly for solving nonlinear problems using the p and h–p versions of the finite element method. The method we discuss in this paper generalizes to hyperbolic initial boundary value problems in multidimensions too, provided the computational boundaries we have introduced are noncharacteristic and the system is symmetrizable. Finally, we show that for the case of analytic coefficients and data, satisfying all the required compatibility conditions so that the solution is analytic, the numerical solution is exponentially accurate in N, where N is proportional to the number of subdomains and the number of degrees of freedom in each element.

20 citations

### Cites background from "A fully spectral method for hyperbo..."

...In fact, it has been shown by Eswaran, Murty and coworkers [10,11] that a purely 8nite di9erence solution may be better than the solution obtained by such a “spectral” scheme when transients are predominant....

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24 Sep 2007

TL;DR: This work considers the numerical solution of an unsteady convection-diffusion equation using high order polynomial approximations both in space and time and enjoys exponential convergence in time and space for analytic solutions.

Abstract: We consider the numerical solution of an unsteady convection-diffusion equation using high order polynomial approximations both in space and time. General boundary conditions and initial conditions can be imposed. The method is fully implicit and enjoys exponential convergence in time and space for analytic solutions. This is conﬁrmed by numerical experiments (in one space dimension) using a spectral element approach in time and a pure spectral method in space. A fast tensor-product solver has been developed to solve the coupled system of algebraic equations for the Ο(N d ) unknown nodal values within a single space-time spectral element. This solver has a ﬁxed complexity of Ο(N d+1 ) ﬂoating point operations and a memory requirement of Ο(N d ) ﬂoating point numbers. An alternative solution method, allowing for a parallel implementation and more easily extendable to more complex problems is sketched out at the end of the paper.

6 citations

### Cites background from "A fully spectral method for hyperbo..."

...Encouraging progress has been made over the past couple of decades [11, 12, 1, 10, 13, 9], however, a significant advance, appropriate for solving real applications, is still missing....

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TL;DR: In this article, a hybrid spectral/finite-difference approach is proposed for the general expansion of the flow field and the solution of the expansion coefficients, where the flow is driven by the movement of a rotating screw; the outer barrel remaining at rest.

Abstract: The three-dimensional Stokes flow in a periodic domain is examined in this study. The problem corresponds closely to the flow inside internal mixers, where the flow is driven by the movement of a rotating screw; the outer barrel remaining at rest. A hybrid spectral/finite-difference approach is proposed for the general expansion of the flow field and the solution of the expansion coefficients. The method is used to determine the flow field between the screw and barrel. The regions of elongation and shear are closely examined. These are the two mechanisms responsible for mixing. Besides its practical importance, the study also allows the assessment of the validity of the various assumptions usually adopted in mixing and lubrication problems. Copyright © 2003 John Wiley & Sons, Ltd.

1 citations

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TL;DR: In this article , a hybrid regularized collision model for advection-diffusion equation is presented, built upon the regularized model, and the coefficients in the reconstruction process of the off-equilibrium distribution are evaluated through two different methods.

Abstract: In this work, a lattice-Boltzmann model for advection-diffusion equation is presented, built upon the regularized model. In the regularized collision model of this paper, the coefficients in the reconstruction process of the off-equilibrium distribution are evaluated through two different methods. The first method arises from direct projection of off-equilibrium distribution, while the second method is computed as to best approximate the scalar diffusion flux. By combining the two methods and introducing proportional coefficients, a hybrid regularized collision model is obtained. In terms of model validations, test cases of smooth problems and discontinuous problems are selected. The results are compared with those obtained by the Bhatnagar–Gross–Krook model and the projection reconstruction model. First, the calculation accuracy of the model in solving the smooth distribution problem is verified by the periodic one-dimensional problem and the advection-diffusion process of the two-dimensional Gaussian distribution. Then, the cases of discontinuous problems are considered for the demonstration of the numerical stability of the model. The results show that the hybrid regularization lattice Boltzmann model retains the advantages of the regularization model in terms of calculation accuracy, and at the same time has good numerical stability in solving discontinuous problems.

##### References

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01 Jan 1987TL;DR: Spectral methods have been widely used in simulation of stability, transition, and turbulence as discussed by the authors, and their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed.

Abstract: Fundamental aspects of spectral methods are introduced. Recent developments in spectral methods are reviewed with an emphasis on collocation techniques. Their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed. The key role that these methods play in the simulation of stability, transition, and turbulence is brought out. A perspective is provided on some of the obstacles that prohibit a wider use of these methods, and how these obstacles are being overcome.

4,632 citations

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TL;DR: In this article, the authors present a set of methods for the estimation of two-dimensional fluid flow, including a Fourier Galerkin method and a Chebyshev Collocation method.

Abstract: 1. Introduction.- 1.1. Historical Background.- 1.2. Some Examples of Spectral Methods.- 1.2.1. A Fourier Galerkin Method for the Wave Equation.- 1.2.2. A Chebyshev Collocation Method for the Heat Equation.- 1.2.3. A Legendre Tau Method for the Poisson Equation.- 1.2.4. Basic Aspects of Galerkin, Tau and Collocation Methods.- 1.3. The Equations of Fluid Dynamics.- 1.3.1. Compressible Navier-Stokes.- 1.3.2. Compressible Euler.- 1.3.3. Compressible Potential.- 1.3.4. Incompressible Flow.- 1.3.5. Boundary Layer.- 1.4. Spectral Accuracy for a Two-Dimensional Fluid Calculation.- 1.5. Three-Dimensional Applications in Fluids.- 2. Spectral Approximation.- 2.1. The Fourier System.- 2.1.1. The Continuous Fourier Expansion.- 2.1.2. The Discrete Fourier Expansion.- 2.1.3. Differentiation.- 2.1.4. The Gibbs Phenomenon.- 2.2. Orthogonal Polynomials in ( - 1, 1).- 2.2.1. Sturm-Liouville Problems.- 2.2.2. Orthogonal Systems of Polynomials.- 2.2.3. Gauss-Type Quadratures and Discrete Polynomial Transforms.- 2.3. Legendre Polynomials.- 2.3.1. Basic Formulas.- 2.3.2. Differentiation.- 2.4. Chebyshev Polynomials.- 2.4.1. Basic Formulas.- 2.4.2. Differentiation.- 2.5. Generalizations.- 2.5.1. Jacobi Polynomials.- 2.5.2. Mapping.- 2.5.3. Semi-Infinite Intervals.- 2.5.4. Infinite Intervals.- 3. Fundamentals of Spectral Methods for PDEs.- 3.1. Spectral Projection of the Burgers Equation.- 3.1.1. Fourier Galerkin.- 3.1.2. Fourier Collocation.- 3.1.3. Chebyshev Tau.- 3.1.4. Chebyshev Collocation.- 3.2. Convolution Sums.- 3.2.1. Pseudospectral Transform Methods.- 3 2 2 Aliasing Removal by Padding or Truncation.- 3.2.3. Aliasing Removal by Phase Shifts.- 3.2.4. Convolution Sums in Chebyshev Methods.- 3.2.5. Relation Between Collocation and Pseudospectral Methods.- 3.3. Boundary Conditions.- 3.4. Coordinate Singularities.- 3.4.1. Polar Coordinates.- 3.4.2. Spherical Polar Coordinates.- 3.5. Two-Dimensional Mapping.- 4. Temporal Discretization.- 4.1. Introduction.- 4.2. The Eigenvalues of Basic Spectral Operators.- 4.2.1. The First-Derivative Operator.- 4.2.2. The Second-Derivative Operator.- 4.3. Some Standard Schemes.- 4.3.1. Multistep Schemes.- 4.3.2. Runge-Kutta Methods.- 4.4. Special Purpose Schemes.- 4.4.1. High Resolution Temporal Schemes.- 4.4.2. Special Integration Techniques.- 4.4.3. Lerat Schemes.- 4.5. Conservation Forms.- 4.6. Aliasing.- 5. Solution Techniques for Implicit Spectral Equations.- 5.1. Direct Methods.- 5.1.1. Fourier Approximations.- 5.1.2. Chebyshev Tau Approximations.- 5.1.3. Schur-Decomposition and Matrix-Diagonalization.- 5.2. Fundamentals of Iterative Methods.- 5.2.1. Richardson Iteration.- 5.2.2. Preconditioning.- 5.2.3. Non-Periodic Problems.- 5.2.4. Finite-Element Preconditioning.- 5.3. Conventional Iterative Methods.- 5.3.1. Descent Methods for Symmetric, Positive-Definite Systems.- 5.3.2. Descent Methods for Non-Symmetric Problems.- 5.3.3. Chebyshev Acceleration.- 5.4. Multidimensional Preconditioning.- 5.4.1. Finite-Difference Solvers.- 5.4.2. Modified Finite-Difference Preconditioners.- 5.5. Spectral Multigrid Methods.- 5.5.1. Model Problem Discussion.- 5.5.2. Two-Dimensional Problems.- 5.5.3. Interpolation Operators.- 5.5.4. Coarse-Grid Operators.- 5.5.5. Relaxation Schemes.- 5.6. A Semi-Implicit Method for the Navier-Stokes Equations.- 6. Simple Incompressible Flows.- 6.1. Burgers Equation.- 6.2. Shear Flow Past a Circle.- 6.3. Boundary-Layer Flows.- 6.4. Linear Stability.- 7. Some Algorithms for Unsteady Navier-Stokes Equations.- 7.1. Introduction.- 7.2. Homogeneous Flows.- 7.2.1. A Spectral Galerkin Solution Technique.- 7.2.2. Treatment of the Nonlinear Terms.- 7.2.3. Refinements.- 7.2.4. Pseudospectral and Collocation Methods.- 7.3. Inhomogeneous Flows.- 7.3.1. Coupled Methods.- 7.3.2. Splitting Methods.- 7.3.3. Galerkin Methods.- 7.3.4. Other Confined Flows.- 7.3.5. Unbounded Flows.- 7.3.6. Aliasing in Transition Calculations.- 7.4. Flows with Multiple Inhomogeneous Directions.- 7.4.1. Choice of Mesh.- 7.4.2. Coupled Methods.- 7.4.3. Splitting Methods.- 7.4.4. Other Methods.- 7.5. Mixed Spectral/Finite-Difference Methods.- 8. Compressible Flow.- 8.1. Introduction.- 8.2. Boundary Conditions for Hyperbolic Problems.- 8.3. Basic Results for Scalar Nonsmooth Problems.- 8.4. Homogeneous Turbulence.- 8.5. Shock-Capturing.- 8.5.1. Potential Flow.- 8.5.2. Ringleb Flow.- 8.5.3. Astrophysical Nozzle.- 8.6. Shock-Fitting.- 8.7. Reacting Flows.- 9. Global Approximation Results.- 9.1. Fourier Approximation.- 9.1.1. Inverse Inequalities for Trigonometric Polynomials.- 9.1.2. Estimates for the Truncation and Best Approximation Errors.- 9.1.3. Estimates for the Interpolation Error.- 9.2. Sturm-Liouville Expansions.- 9.2.1. Regular Sturm-Liouville Problems.- 9.2.2. Singular Sturm-Liouville Problems.- 9.3. Discrete Norms.- 9.4. Legendre Approximations.- 9.4.1. Inverse Inequalities for Algebraic Polynomials.- 9.4.2. Estimates for the Truncation and Best Approximation Errors.- 9.4.3. Estimates for the Interpolation Error.- 9.5. Chebyshev Approximations.- 9.5.1. Inverse Inequalities for Polynomials.- 9.5.2. Estimates for the Truncation and Best Approximation Errors.- 9.5.3. Estimates for the Interpolation Error.- 9.5.4. Proofs of Some Approximation Results.- 9.6. Other Polynomial Approximations.- 9.6.1. Jacobi Polynomials.- 9.6.2. Laguerre and Hermite Polynomials.- 9.7. Approximation Results in Several Dimensions.- 9.7.1. Fourier Approximations.- 9.7.2. Legendre Approximations.- 9.7.3. Chebyshev Approximations.- 9.7.4. Blended Fourier and Chebyshev Approximations.- 10. Theory of Stability and Convergence for Spectral Methods.- 10.1. The Three Examples Revisited.- 10.1.1. A Fourier Galerkin Method for the Wave Equation.- 10.1.2. A Chebyshev Collocation Method for the Heat Equation.- 10.1.3. A Legendre Tau Method for the Poisson Equation.- 10.2. Towards a General Theory.- 10.3. General Formulation of Spectral Approximations to Linear Steady Problems.- 10.4. Galerkin, Collocation and Tau Methods.- 10.4.1. Galerkin Methods.- 10.4.2. Tau Methods.- 10.4.3. Collocation Methods.- 10.5. General Formulation of Spectral Approximations to Linear Evolution Equations.- 10.5.1. Conditions for Stability and Convergence: The Parabolic Case.- 10.5.2. Conditions for Stability and Convergence: The Hyperbolic Case.- 10.6. The Error Equation.- 11. Steady, Smooth Problems.- 11.1. The Poisson Equation.- 11.1.1. Legendre Methods.- 11.1.2. Chebyshev Methods.- 11.1.3. Other Boundary Value Problems.- 11.2. Advection-Diffusion Equation.- 11.2.1. Linear Advection-Diffusion Equation.- 11.2.2. Steady Burgers Equation.- 11.3. Navier-Stokes Equations.- 11.3.1. Compatibility Conditions Between Velocity and Pressure.- 11.3.2. Direct Discretization of the Continuity Equation: The \"inf-sup\" Condition.- 11.3.3. Discretizations of the Continuity Equation by an Influence-Matrix Technique: The Kleiser-Schumann Method.- 11.3.4. Navier-Stokes Equations in Streamfunction Formulation.- 11.4. The Eigenvalues of Some Spectral Operators.- 11.4.1. The Discrete Eigenvalues for Lu = ? uxx.- 11.4.2. The Discrete Eigenvalues for Lu = ? vuxx + bux.- 11.4.3. The Discrete Eigenvalues for Lu = ux.- 12. Transient, Smooth Problems.- 12.1. Linear Hyperbolic Equations.- 12.1.1. Periodic Boundary Conditions.- 12.1.2. Non-Periodic Boundary Conditions.- 12.1.3. Hyperbolic Systems.- 12.1.4. Spectral Accuracy for Non-Smooth Solutions.- 12.2. Heat Equation.- 12.2.1. Semi-Discrete Approximation.- 12.2.2. Fully Discrete Approximation.- 12.3. Advection-Diffusion Equation.- 12.3.1. Semi-Discrete Approximation.- 12.3.2. Fully Discrete Approximation.- 13. Domain Decomposition Methods.- 13.1. Introduction.- 13.2. Patching Methods.- 13.2.1. Notation.- 13.2.2. Discretization.- 13.2.3. Solution Techniques.- 13.2.4. Examples.- 13.3. Variational Methods.- 13.3.1. Formulation.- 13.3.2. The Spectral-Element Method.- 13.4. The Alternating Schwarz Method.- 13.5. Mathematical Aspects of Domain Decomposition Methods.- 13.5.1. Patching Methods.- 13.5.2. Equivalence Between Patching and Variational Methods.- 13.6. Some Stability and Convergence Results.- 13.6.1. Patching Methods.- 13.6.2. Variational Methods.- Appendices.- A. Basic Mathematical Concepts.- B. Fast Fourier Transforms.- C. Jacobi-Gauss-Lobatto Roots.- References.

3,753 citations

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TL;DR: The algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution.

Abstract: A pseudospectral numerical scheme for solving linear, periodic, hyperbolic problems is described. It has infinite accuracy both in time and in space. The high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular, one step explicit scheme. The algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution. The class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator.

164 citations

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TL;DR: In this paper, a spectral method for initial boundary value problems is presented, where a filtered version of the partial differential equation and the initial and boundary conditions at an overdetermined set of points are collocated.

Abstract: This paper presents a new approach to spectral methods for initial boundary value problems. A filtered version of the partial differential equation and the initial and boundary conditions at an overdetermined set of points are collocated. As an approximate solution, the function is chosen that belongs to an appropriate finite-dimensional space and minimizes a weighted average of the residuals at these points. It is proved that the approximate solution converges to the actual solution at a spectral rate of accuracy in both space and time. The proof is based on a priori energy estimates that have been proved for such systems. Although this method is restricted here to hyperbolic initial boundary value problems and Chebyshev polynomials, it generalizes to general initial boundary value problems, boundary value problems, and Gegenbauer polynomials.

22 citations

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TL;DR: In this paper, a spectral method based on the Chebyshev collocation technique and finite-difference time stepping was proposed for solving the one-dimensional shallowwater wave equations.

Abstract: 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.

10 citations