Chebyshev Tau Spectral Method Research Articles

A spectral method for the depth-separated solution of a wavenumber integration model for horizontally stratified fluid acoustic waveguides

The wavenumber integration model is the most precise approach for assessing arbitrary horizontally stratified media within the sphere of computational ocean acoustics. Unlike the normal-mode approach, it considers not only discrete spectra but also continuous spectral components, resulting in fewer model approximation errors for horizontally stratified media. Traditionally, the depth-separated wave equation in the wavenumber integration model has been solved using analytical and semianalytical methods, and numerical solutions have been primarily based on the finite difference and finite element methods. This paper proposes an algorithm for solving the depth equation via the Chebyshev–Tau spectral method, combined with a domain decomposition strategy, resulting in the development of a numerical program named WISpec. The algorithm can simulate the sound field excitation not only from a point source but also from an infinite line source. To that end, the depth equations for each layer are first discretized through the Chebyshev–Tau spectral method and subsequently solved simultaneously by incorporating boundary and interface conditions. Representative numerical experiments are presented to validate the accuracy and speed of WISpec. The high degree of consistency of results obtained from different software tools running the same configuration provides ample evidence that the numerical algorithm described in this paper is accurate, reliable, and numerically stable.

A Chebyshev–Tau spectral method for coupled modes of underwater sound propagation in range-dependent ocean environments

A coupled-mode model is a classic approach for solving range-dependent sound propagations and is often used to provide benchmark solutions in comparison with other numerical models because of its high accuracy. Existing coupled-mode programs have disadvantages such as high computational cost, weak adaptability to complex ocean environments, and numerical instability. In this paper, a new algorithm that uses an improved range normalization of a “stair-step” and global matrix approach to address range dependence in ocean environments is designed. This algorithm uses the Chebyshev–Tau spectral method to solve the eigenpairs in the range-independent segments. The Chebyshev–Tau spectral method can converge rapidly, and the rate of convergence depends on the smoothness of the sound speed and density profiles. The main steps of the algorithm are parallelized, so parallel computing technologies are also applied for further acceleration. Based on this algorithm, an efficient program is implemented, and numerical simulations verify that this algorithm is reliable, accurate, and capable. Compared with the existing coupled-mode programs, the newly developed program is more stable and efficient with comparable accuracy and can simulate waveguides in more complex and realistic ocean environments.

Application of a spectral method to simulate quasi-three-dimensional underwater acoustic fields

The calculation of a three-dimensional underwater acoustic field has always been a key problem in computational ocean acoustics. Traditionally, this solution is usually obtained by directly solving the acoustic Helmholtz equation using a finite difference or finite element algorithm. Solving the three-dimensional Helmholtz equation directly is computationally expensive. For quasi-three-dimensional problems, the Helmholtz equation can be processed by the integral transformation approach, which can greatly reduce the computational cost. In this paper, a numerical algorithm for a quasi-three-dimensional sound field that combines an integral transformation technique, stepwise coupled modes and a spectral method is designed. The quasi-three-dimensional problem is transformed into a two-dimensional problem using an integral transformation strategy. A stepwise approximation is then used to discretize the range dependence of the two-dimensional problem; this approximation is essentially a physical discretization that further reduces the range-dependent two-dimensional problem to a one-dimensional problem. Finally, the Chebyshev–Tau spectral method is employed to accurately solve the one-dimensional problem. We provide the corresponding numerical program SPEC3D for the proposed algorithm and describe several representative numerical examples. In the numerical experiments, the consistency between SPEC3D and the analytical solution/high-precision finite difference program COACH verifies the reliability and capability of the proposed algorithm. A comparison of running times illustrates that the algorithm proposed in this paper is significantly faster than the full three-dimensional algorithm in terms of computational speed.

A novel algorithm to solve for an underwater line source sound field based on coupled modes and a spectral method

A high-precision numerical sound field is the basis of underwater target detection, positioning and communication. A line source in a plane is a common type of sound source in computational ocean acoustics. The exciting waveguide in a range-dependent ocean environment is often structurally complicated; however, traditional algorithms often assume that the waveguide has a simple seabed boundary and that the line source is located at a horizontal range of 0 m, although this ideal situation is rarely encountered in the actual ocean. In this paper, a novel algorithm is designed that can solve for the sound field excited by a line source at any position in a range-dependent ocean environment. The proposed algorithm uses the classic stepwise approximation approach to address the range dependence of the environment and uses the Chebyshev–Tau spectral method to solve for the horizontal wavenumbers and modes of approximately range-independent segments. Once the modal information of these flat segments has been obtained, a global matrix is constructed to solve for the coupling coefficients of all segments, and finally, the complete sound field is synthesized. Numerical experiments using a robust numerical program developed based on this algorithm verify the correctness and usability of our novel algorithm and software. Furthermore, a detailed analysis and test of the computational cost of this algorithm show that it is efficient.

Applying the Chebyshev–Tau Spectral Method to Solve the Parabolic Equation Model of Wide-Angle Rational Approximation in Ocean Acoustics

Solving an acoustic wave equation using a parabolic approximation is a popular approach for many existing ocean acoustic models. Commonly used parabolic equation (PE) model programs, such as the range-dependent acoustic model (RAM), are discretized by the finite difference method (FDM). Considering the idea and theory of the wide-angle rational approximation, a discrete PE model using the Chebyshev spectral method (CSM) is derived, and the code is developed. This method is currently suitable only for range-independent waveguides. Taking three ideal fluid waveguides as examples, the correctness of using the CSM discrete PE model in solving the underwater acoustic propagation problem is verified. The test results show that compared with the RAM, the method proposed in this paper can achieve higher accuracy in computational underwater acoustics and requires fewer discrete grid points. After optimization, this method is more advantageous than the FDM in terms of speed. Thus, the CSM provides high-precision reference standards for benchmark examples of the range-independent PE model.

Application of a Chebyshev Collocation Method to Solve a Parabolic Equation Model of Underwater Acoustic Propagation

The parabolic approximation has been used extensively for underwater acoustic propagation and is attractive because it is computationally efficient. Widely used parabolic equation (PE) model programs such as the range-dependent acoustic model (RAM) are discretized by the finite difference method. Based on the idea of the Pad $$\acute{\text {e}}$$ series expansion of the depth operator, a new discrete PE model using the Chebyshev collocation method (CCM) is derived, and the code (CCMPE) is developed. Taking the problems of four ideal fluid waveguides as experiments, the correctness of the discrete PE model using the CCM to solve a simple underwater acoustic propagation problem is verified. The test results show that the CCMPE developed in this article achieves higher accuracy in the calculation of underwater acoustic propagation in a simple marine environment and requires fewer discrete grid points than the finite difference discrete PE model. Furthermore, although the running time of the proposed method is longer than that of the finite difference discrete PE program (RAM), it is shorter than that of the Chebyshev–Tau spectral method.

A Chebyshev-Tau spectral method for normal modes of underwater sound propagation with a layered marine environment

The normal mode model is one of the most popular approaches for solving underwater sound propagation problems. Among other methods, the finite difference method is widely used in classic normal mode programs. In many recent studies, the spectral method has been used for discretization. It is generally more accurate than the finite difference method. However, the spectral method requires that the variables to be solved are continuous in space, and the traditional spectral method is powerless for a layered marine environment. A Chebyshev-Tau spectral method based on domain decomposition is applied to the construction of underwater acoustic normal modes in this paper. In this method, the differential equation is projected onto spectral space from the original physical space with the help of an orthogonal basis of Chebyshev polynomials. A complex matrix eigenvalue / eigenvector problem is thus formed, from which the solution of horizontal wavenumbers and modal functions can be solved. The validity of the acoustic field calculation is tested in comparison with classic programs. The results of analysis and tests show that compared with the classic finite difference method, the proposed Chebyshev-Tau spectral method has the advantage of high computational accuracy. In addition, in terms of running time, our method is faster than the Legendre-Galerkin spectral method.

Stability of falling liquid films on flexible substrates

Abstract

A third-order multistep time discretization for a Chebyshev tau spectral method

A time discretization scheme based on the third-order backward difference formula has been embedded into a Chebyshev tau spectral method for the Navier-Stokes equations. The time discretization is a variant of the second-order backward scheme proposed by Krasnov et al. (2008) 3. High-resolution direct numerical simulations of turbulent incompressible channel flow have been performed to compare the backward scheme to the Runge-Kutta scheme proposed by Spalart et al. (1991) 2. It is shown that the Runge-Kutta scheme leads to a poor convergence of some third-order spatial derivatives in the direct vicinity of the wall, derivatives that represent the diffusion of wall-tangential vorticity. The convergence at the wall is shown to be significantly improved if the backward scheme is applied.

An efficient Chebyshev–Tau spectral method for Ginzburg–Landau–Schrödinger equations

We propose an efficient time-splitting Chebyshev–Tau spectral method for the Ginzburg–Landau–Schrödinger equation with zero/nonzero far-field boundary conditions. The key technique that we apply is splitting the Ginzburg–Landau–Schrödinger equation in time into two parts, a nonlinear equation and a linear equation. The nonlinear equation is solved exactly; while the linear equation in one dimension is solved with Chebyshev–Tau spectral discretization in space and Crank–Nicolson method in time. The associated discretized system can be solved very efficiently since they can be decoupled into two systems, one for the odd coefficients, the other for the even coefficients. The associated matrices have a quasi-tridiagonal structure which allows a direction solution to be obtained. The computation cost of the method in one dimension is O ( N log ( N ) ) compared with that of the non-optimized one, which is O ( N 2 ) . By applying the alternating direction implicit (ADI) technique, we extend this efficient method to solve the Ginzburg–Landau–Schrödinger equation both in two dimensions and in three dimensions, respectively. Numerical accuracy tests of the method in one dimension, two dimensions and three dimensions are presented. Application of the method to study the semi-classical limits of Ginzburg–Landau–Schrödinger equation in one dimension and the two-dimensional quantized vortex dynamics in the Ginzburg–Landau–Schrödinger equation are also presented.

The Chebyshev Tau Spectral Method for the Solution of the Linear Stability Equations for Rayleigh-Bénard Convection with Melting

The Chebyshev Tau Spectral Method for the Solution of the Linear Stability Equations for Rayleigh-Bénard Convection with Melting

Analytical and Numerical Investigation of the Resolvent for Plane Couette Flow

We present new bounds for the solution of the resolvent equation for plane Couette flow. Both analytic methods and computation, using the Chebyshev tau spectral method, are used. The emphasis is on determining the Reynolds number-dependence of the estimates. The main result is the introduction of a weighted norm, which leads to optimal asymptotic behavior of the resolvent for large Reynolds numbers.

Simulation of dissolution of silicon in an indium solution by spectral methods

The results of a numerical simulation of natural convection due to concentration gradients during dissolution of silicon in an indium solution in a horizontal substrate–solution–substrate system are presented. The Chebyshev–Tau spectral method has been used for the simulations. The results are in very good agreement with the experimental data available in the literature. It is concluded that the discrepancies in the dissolution depths between the previous simulations and experimental data, especially at the earlier stages of the process, are mostly due to combined effects of uncertainties in the predicted properties and insufficient numerical accuracy.

Stability analysis of constitutive equations for polymer melts in viscometric flows

The stability of two representative constitutive equations for polymer melts, the exponential Phan-Thien–Tanner (PTT) model and the Giesekus model, are investigated in planar shear flows. For the PTT equation, instabilities are predicted for both plane Couette and Poiseuille flows using transient finite-element calculations. A Chebyshev–Tau spectral method is used to confirm that these instabilities are not spurious or an artifact of the finite element formulation. Mechanisms are proposed based on an energy analysis of the most unstable mode for each flow. The stability of plane Couette flow of a Giesekus model is also probed using our spectral method and found to be stable for the range of parameters investigated. However, in pressure driven flow, the Giesekus model is unstable over a critical local Weissenberg number ( Wi) based on the shear rate at the channel wall. We present the complete eigenspectrum for this model in both Couette and Poiseuille flows and note that ‘ballooning’ of the continuous spectrum, which can cause spurious instabilities, is significantly stabilized for this constitutive equation relative to the upper convected Maxwell model or PTT model. Both Poiseuille flow instabilities occur at moderate Weissenberg numbers and, with more careful investigation, may be able to explain some of the unusual phenomena observed in slit flows of polymer melts [AIChE J. 22 (2) (1976) 209; J. Non-Newt. Fluid Mech. 77 (1998) 123; Adv. Polym. Sci. 138 (1999) 227].

The Origin and Nature of Spurious Eigenvalues in the Spectral Tau Method

The Chebyshev–tau spectral method for approximating eigenvalues of boundary value problems may produce spurious eigenvalues with large positive real parts, even when all true eigenvalues of the problem are known to have negative real parts. We explain the origin and nature of the “spurious eigenvalues” in an example problem. The explanation will demonstrate that the large positive eigenvalues are an approximation of infinite eigenvalues in a nearby generalized eigenvalue problem.

Spanwise pairing of finite-amplitude longitudinal vortex rolls in inclined free-convection boundary layers

Buoyancy-driven flow on a heated inclined plate can become unstable to static longitudinal roil instability at a critical distance, measured by $\tilde{R}_{\rm c}$, from the leading edge. Experiments in water by Sparrow & Husar (1969) indicate that these rolls undergo a second transition further downstream such that adjacent rolls merge and their spanwise wavelength is doubled. We study this secondary bifurcation phenomenon here with a set of model equations by first constructing the full eigenspectrum and eigenfunctions with a Chebyshev–Tau spectral method and then deriving the pertinent amplitude equations. By stipulating that the dimensional cross-stream wavelength of the rolls remains constant beyond $\tilde{R}_{\rm c}$, which is consistent with experimental observation, we show that the finite-amplitude primary rolls are destabilized by the ½ subharmonic mode at another critical distance $\tilde{R}_{\frac{1}{2}}$ from the edge. This ½ mode is shown to have an asymmetric spatial phase shift of ½π relative to the original 1 mode of the primary rolls, thus explaining the unique dislocation of tracer streaks after the rolls coalesce in the experiments. Also consistent with experimental observation is the theoretical result that the merged rolls are annihilated downstream by a saddle-node bifurcation before further wavclength doubling can occur. Simple amplitude criteria and critical distances from the leading edge for the various transitions are derived and compared to experimental values.

Elimination of spurious eigenvalues in the Chebyshev tau spectral method

Spectral methods have been used to great advantage in hydrodynamic stability calculations; the concepts are described in Orszag's seminal application of the Chebyshev tau method to the Orr-Sommerfeld equation for plane Poiseuille flow in 1971. Orszag discusses both the Chebyshev Galerkin and the Chebyshev tau methods, but presents results for the tau method, which is easier to implement than the Galerkin method. The tau method has the disadvantage that two unstable eigenvalues are produced that are artifacts of the discretization. An extremely simple modification to the Chebyshev tau method is presented which eliminates the spurious eigenvalues. First a simplified model of the Orr-Sommerfeld equation discussed by Gottlieb and Orszag was studied. Then the Chebyshev tau method is considered, which has two spurious eigenvalues, and then a modification which eliminates them is described. Finally, results for the Orr-Sommerfeld equation are considered where the modified tau method also eliminates the spurious eigenvalues. The simplicity of the modification makes it a convenient alternative to other approaches to the problem.