The extended Fourier pseudospectral time-domain method for atmospheric sound propagation

An extended Fourier pseudospectral time-domain (PSTD) method is presented to model atmospheric sound propagation by solving the linearized Euler equations. In this method, evaluation of spatial derivatives is based on an eigenfunction expansion. Evaluation on a spatial grid requires only two spatial points per wavelength. Time iteration is done using a low-storage optimized six-stage Runge-Kutta method. This method is applied to two-dimensional non-moving media models, one with screens and one for an urban canyon, with generally high accuracy in both amplitude and phase. For a moving atmosphere, accurate results have been obtained in models with both a uniform and a logarithmic wind velocity profile over a rigid ground surface and in the presence of a screen. The method has also been validated for three-dimensional sound propagation over a screen. For that application, the developed method is in the order of 100 times faster than the second-order-accurate FDTD solution to the linearized Euler equations. The method is found to be well suited for atmospheric sound propagation simulations where effects of complex meteorology and straight rigid boundary surfaces are to be investigated.     

Reliable reduced-order models for time-dependent linearized Euler equations

Development of optimal reduced-order models for linearized Euler equations is investigated. Recent methods based on proper orthogonal decomposition (POD), applicable for high-order systems, are presented and compared. Particular attention is paid to the link between the choice of the projection and the efficiency of the reduced model. A stabilizing projection is introduced to induce a stable reduced-order model at finite time even if the energy of the physical model is growing. The proposed method is particularly well adapted for time-dependent hyperbolic systems and intrinsically skew-symmetric models. This paper also provides a common methodology to reliably reduce very large nonsymmetric physical problems.     

A pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations

A new Chebyshev pseudospectral technique (based on the projection method that was previously applied by the authors to the solution of two-dimensional incompressible Navier-Stokes equations in primitive variables for nonperiodic boundary conditions) is extended to solve the three-dimensional Navier-Stokes equations. The crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain, on the boundaries as well as in the interior. The key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure Poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition. The method was tested on a two-dimensional driven cavity flow and the results were compared with those of the most accurate finite-difference calculation. The three-dimensional driven cavity flow was then calculated at the same Reynolds numbers as the two-dimensional cases, i.e., Re = 100, 400, and 1000. In the calculated results, three-dimensional boundary effects were observed in all cases and became more apparent with increasing Reynolds number.     

Semi-empirical boundary conditions for the linearized acoustic Euler equations using Pseudo-Spectral Time-Domain methods

Pseudo-Spectral Time-Domain algorithms have emerged as new numerical methods for solving Eulerian problems. These methods, in contrast to more common finite-difference, time-domain approaches, provide isotropic dispersion characteristics. However, the technical literature concerning to this topic presents a serious lack of methods for dealing with partially reflecting boundary conditions in order to simulate surfaces of a specified impedance. In the current paper we present a novel semi-empirical formulation for simulating constant impedance boundary conditions within Pseudo-Spectral techniques based on the Fourier transform. Finally, the validations in one and two dimensions by means of different numerical experiments, show the accuracy of the model.     

Large time behavior and asymptotic stability of the 2D Euler and linearized Euler equations

We study the asymptotic behavior and the asymptotic stability of the 2D Euler equations and of the 2D linearized Euler equations close to parallel flows. We focus on flows with spectrally stable profiles U(y) and with stationary streamlines y=y₀ (such that U^′(y₀)=0), a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of this ensemble of flow profiles even in the absence of any dissipative mechanisms.     

An analytical derivation of the optimum source patterns for the pseudospectral time-domain method

In the computational electromagnetics and acoustics, spatially smoothed sources are often utilized to alleviate the aliasing errors in the pseudospectral time-domain (PSTD) algorithms. In our work, an analytical derivation of the optimum source patterns is presented according to the accurately derived expressions of the dominant source-introduced aliasing errors according to the circular discrete convolution and Tailor series expansion method. We quantitatively demonstrate, for the first time in literature, that the aliasing errors can be optimally suppressed and rapidly reduced to the negligible levels by these optimum patterns and with the increment of source cells. We also provide the different implementation schemes of the optimal patterns both for the soft and hard source cases. The numerical calculation and 1D PSTD transient simulations are conducted to verify the excellent performance of these optimum sources.     

Variational iteration method for solving compressible Euler equations

This paper applies the variational iteration method to obtain approximate analytic solutions of compressible Euler equations in gas dynamics.This method is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional.Using this method,a rapid convergent sequence is produced which converges to the exact solutions of the problem.Numerical results and comparison with other two numerical solutions verify that this method is very convenient and efficient.     

On the unstable discrete spectrum of the linearized 2-D Euler equations in bounded domains

We investigate the behavior of the unstable discrete spectrum of the linearized 2-D Euler equation when the domain is smoothly perturbed. It is shown that when a self-adjoint Schrödinger-type operator undergoes a codimension-1 bifurcation it translates into a bifurcation in the linearized Euler equation associated with an instability either appearing or disappearing.We give sufficient conditions in order to observe smooth quadratic growth of the unstable eigencurves of the linearized Euler equation. The critical exponent is explicitly given as a function of the null-vector involved into the codimension-1 bifurcation using first and second-order moments of a Laplace transform.This analysis provides an explanation for the successive symmetry-breaking bifurcations observed in models of the mid-latitude oceans. An explicit example is also given.     

电磁场2.5维时域间断有限元方法

介质沿空间固定方向均匀分布的结构在电磁导波器件中有十分广泛的应用,对这类器件的分析通常被称为2.5D电磁问题。利用器件在固定方向介质分布均匀的特点,将电磁场量沿该方向进行空间傅里叶变换,可以把对三维问题的分析转化为两维问题求解,从而极大地减小计算开销。针对传统基于差分的2.5D电磁场算法在弯曲形状逼近上有阶梯误差的缺陷,本文提出了基于三角形网格的2.5D时域间断有限元方法（DGTD）,并用它模拟了电偶极子与光纤的耦合效率和光子晶体光纤的色散特性。与基于规则网格的2.5D差分方法进行对比。结果表明,文中建立的2.5D DGTD方法对弯曲形状的模拟更加逼真,计算内存占用最大减少10.4%,计算精度最大相差0.011%,计算时间缩短74.9%,计算效率提高。     

A conservative Fourier pseudospectral algorithm for a coupled nonlinear Schroedinger system

We derive a new method for a coupled nonlinear Schr/Sdinger system by using the square of first-order Fourier spectral differentiation matrix D1 instead of traditional second-order Fourier spectral differentiation matrix D2 to approximate the second derivative. We prove the proposed method preserves the charge and energy conservation laws exactly. In numerical tests, we display the accuracy of numerical solution and the role of the nonlinear coupling parameter in cases of soliton collisions. Numerical experiments also exhibit the excellent performance of the method in preserving the charge and energy conservation laws. These numerical results verify that the proposed method is both a charge-preserving and an energy-preserving algorithm.     

A Hermite WENO reconstruction-based discontinuous Galerkin method for the Euler equations on tetrahedral grids

A Hermite WENO reconstruction-based discontinuous Galerkin method RDG(P₁P₂), designed not only to enhance the accuracy of discontinuous Galerkin method but also to ensure linear stability of the RDG method, is presented for solving the compressible Euler equations on tetrahedral grids. In this RDG(P₁P₂) method, a quadratic polynomial solution (P₂) is first reconstructed using a least-squares method from the underlying linear polynomial (P₁) discontinuous Galerkin solution. By taking advantage of handily available and yet invaluable information, namely the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact and consistent with the underlying DG method. The final quadratic polynomial solution is then obtained using a WENO reconstruction, which is necessary to ensure linear stability of the RDG method. The developed RDG method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical experiments demonstrate that the developed RDG(P₁P₂) method is able to maintain the linear stability, achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method without significant increase in computing costs and storage requirements.     

A hybrid multilevel method for high-order discretization of the Euler equations on unstructured meshes

Higher order discretization has not been widely successful in industrial applications to compressible flow simulation. Among several reasons for this, one may identify the lack of tailor-suited, best-practice relaxation techniques that compare favorably to highly tuned lower order methods, such as finite-volume schemes. In this paper we investigate solution algorithms in conjunction with high-order Spectral Difference discretization for the Euler equations, using such techniques as multigrid and matrix-free implicit relaxation methods. In particular we present a novel hybrid multilevel relaxation method that combines (optionally matrix-free) implicit relaxation techniques with explicit multistage smoothing using geometric multigrid. Furthermore, we discuss efficient implementation of these concepts using such tools as automatic differentiation.     

The discontinuous Petrov-Galerkin method for one-dimensional compressible Euler equations in the Lagrangian coordinate

In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume (RKCV) discontinuous finite element method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discontinuous Galerkin (RKDG) method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.     

A high-order multi-dimensional HLL-Riemann solver for non-linear Euler equations

This article presents a numerical model that enables to solve on unstructured triangular meshes and with a high-order of accuracy, a multi-dimensional Riemann problem that appears when solving hyperbolic problems.For this purpose, we use a MUSCL-like procedure in a “cell-vertex” finite-volume framework. In the first part of this procedure, we devise a four-state bi-dimensional HLL solver (HLL-2D). This solver is based upon the Riemann problem generated at the centre of gravity of a triangular cell, from surrounding cell-averages. A new three-wave model makes it possible to solve this problem, approximately. A first-order version of the bi-dimensional Riemann solver is then generated for discretizing the full compressible Euler equations.In the second part of the MUSCL procedure, we develop a polynomial reconstruction that uses all the surrounding numerical data of a given point, to give at best third-order accuracy. The resulting over determined system is solved by using a least-square methodology. To enforce monotonicity conditions into the polynomial interpolation, we develop a simplified central WENO (CWENO) procedure.Numerical tests and comparisons with competing numerical methods enable to identify the salient features of the whole model.     

The EPS method: A new method for constructing pseudospectral derivative operators

We develop a new type of derivative matrix for pseudospectral methods. The norm of these matrices grows at the optimal rate O(N²) for N-by-N matrices, in contrast to standard pseudospectral constructions that result in O(N⁴) growth of the norm. The smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation. The construction is based on representing the derivative operator as an integral kernel, and does not rely on the interpolating polynomials. In particular, we construct second derivative matrices that incorporate Dirichlet or Neumann boundary conditions on an interval and on the disk, but the method can be used to construct a wide variety of commonly used operators for solving PDEs and integral equations. The construction can be used with any quadrature, including traditional Gauss–Legendre quadratures, but we have found that by using quadratures based on prolate spheroidal wave functions, we can achieve a near optimal sampling rate close to two points per wavelength, even for non-periodic problems. We provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices, while resulting in a norm that is orders of magnitude smaller than the standard construction. To demonstrate the advantage of the new construction, we apply the method for solving the wave equation in constant and discontinuous media and for solving PDEs on the unit disk. We also present two compression algorithms for applying the derivative matrices in O(N log N) operations.     

Analytic Hessian derivation for the quasi-one-dimensional Euler equations

The Hessian for the quasi-one-dimensional Euler equations is derived. A pressure minimization problem and a pressure matching inverse problem are considered. The flow sensitivity, adjoint sensitivity, gradient and Hessian are calculated analytically using a direct approach that is specific to the model problems. For the pressure minimization problem we find that the Hessian exists and it contains elements with significantly larger values around the shock location. For the pressure matching inverse problem we find at least one case for which the gradient as well as the Hessian do not exist. In addition, two formulations for calculating the Hessian are proposed and implemented for the given problems. Both methods can be implemented in industrial applications such as large scale aerodynamic optimization.     

Second order front tracking for the Euler equations

A second order front tracking method is developed for solving the hyperbolic system of Euler equations of inviscid fluid dynamics numerically. Meshless front tracking methods are usually limited to first order accuracy, since they are based on a piecewise constant approximation of the solution. Here second order convergence is achieved by deriving a piecewise linear reconstruction of the piecewise constant front tracking solution. The linearization is performed by decomposing the front tracking solution into its wave components and by linearizing the wave solutions separately. In order to construct a physically correct linearization, the physical phenomena of the front are taken into account in terms of the front types of the previously developed improved front interaction model. This front interaction model is also extended to include front numbers used in the wave decomposition. It is illustrated numerically for Sod’s Riemann problem, the two interacting blast waves problem, and a two-dimensional supersonic airfoil flow validation study that the proposed front tracking method achieves second order convergence also in the presence of strong discontinuities and their interactions.     

A multislope MUSCL method on unstructured meshes applied to compressible Euler equations for axisymmetric swirling flows

A finite volume method for the numerical solution of axisymmetric inviscid swirling flows is presented. The governing equations of the flow are the axisymmetric compressible Euler equations including swirl (or tangential) velocity. A first-order scheme is introduced where the convective fluxes at cell interfaces are evaluated by the Rusanov or the HLLC numerical flux while the geometric source terms are discretizated to provide a well-balanced scheme i.e. the steady-state solutions with null velocity are preserved. Extension to the second-order space approximation using a multislope MUSCL method is then derived. To test the numerical scheme, a stationary solution of the fluid flow following the radial direction has been established with a zero and nonzero tangential velocity. Numerical and exact solutions are compared for classical Riemann problems where we employ different limiters and effectiveness of the multislope MUSCL scheme is demonstrated for strongly shocked axially symmetric flows like in spherical bubble compression problem. Two other tests with axisymmetric geometries are performed: the supersonic flow in a tube with a cone and the axisymmetric blunt body with a free stream.     

A new class of gas-kinetic relaxation schemes for the compressible Euler equations

Starting from the gas-kinetic model, a new class of relaxation schemes for the Euler equations is presented. In contrast to the Riemann solver, these schemes provide a multidimensional dynamical gas evolution model, which combines both Lax-Wendroff and kinetic flux vector splitting schemes, and their coupling is based on the fact that a nonequilibrium state will evolve into an equilibrium state along with the increase of entropy. The numerical fluxes are constructed without getting into the details of the particle collisions. The results for many well-defined test cases are presented to indicate the robustness and accuracy of the current scheme.