Pseudospectral Numerical Scheme Research Articles

Hydrogen-oxygen flame acceleration in narrow open ended channels

The problem of sudden flame acceleration in narrow channels and subsequent Deflagration to Detonation Transition (DDT) is revisited. The hydrogen–oxygen combustion system is considered both experimentally and numerically. The flame is initiated at one open end of a square narrow channel of different widths (4, 8, 10 and 20 mm) and propagates to another open end. Experimental results show a high sensitivity to the mixture composition: for higher mixture reactivity, an abrupt flame acceleration is reported. In 4 × 4 mm2 channel, when the mixture composition is close to stoichiometry (2:1), the DDT is observed without any evidence of shock waves prior to the detonation transition. Two complementary sub-models are combined to account for the effect of walls on flow and on flame front geometry. A pseudo-spectral numerical scheme is used to integrate the system of equations both in time and in space. The simulation results show how the suggested model reproduces main features of the phenomenon. A simple criterion is suggested for the onset of the flame self-acceleration. There exists a critical flame folding factor for the DDT, which is an invariant of the system. The reported critical folding factor is compared to the theoretical estimation introduced by Gordon et al (2020).

A second order accuracy in time, Fourier pseudo-spectral numerical scheme for "Good" Boussinesq equation

The nonlinear stability and convergence of a numerical scheme for the 'Good' Boussinesq equation is provided in this article, with second order temporal accuracy and Fourier pseudo-spectral approximation in space. Instead of introducing an intermediate variable $ \psi $ to approximate the first order temporal derivative, we apply a direct approximation to the second order temporal derivative, which in turn leads to a reduction of the intermediate numerical variable and improvement in computational efficiency. A careful analysis reveals an unconditional stability and convergence for such a temporal discretization. In addition, by making use of the techniques of aliasing error control, we obtain an $ \ell^\infty (0,T^*; H^2) $ convergence for $ u $ and $ \ell^\infty (0,T^*; \ell^2) $ convergence for the discrete time-derivative of the solution in this paper, in comparison with the $ \ell^\infty (0,T^*; \ell^2) $ convergence for $ u $ and the $ \ell^\infty (0,T^*; H^{-2}) $ convergence for the time-derivative, given in [19].

Optimization via Chebyshev polynomials

This paper presents for the first time a robust exact line-search method based on a full pseudospectral (PS) numerical scheme employing orthogonal polynomials. The proposed method takes on an adaptive search procedure and combines the superior accuracy of Chebyshev PS approximations with the high-order approximations obtained through Chebyshev PS differentiation matrices (CPSDMs). In addition, the method exhibits quadratic convergence rate by enforcing an adaptive Newton search iterative scheme. A rigorous error analysis of the proposed method is presented along with a detailed set of pseudocodes for the established computational algorithms. Several numerical experiments are conducted on one- and multi-dimensional optimization test problems to illustrate the advantages of the proposed strategy.

High accuracy numerical methods for the Gardner–Ostrovsky equation

The Gardner–Ostrovsky equation, also known as the extended rotation-modified Korteweg–de Vries (KdV) equation, describes weakly nonlinear internal oceanic waves under the influence of Earth’ rotation. High accuracy numerical methods are needed to follow with precision the long time evolution of the solutions of this equation, with the additional difficulty that the numerical methods have to conserve very accurately several invariants of the solutions, including the mass and the energy of the waves. Finite-difference methods traditionally used for the solution of the KdV equation fails to preserve accurately these invariants for large times. In this paper we show that this difficulty can be overcome by using a high accuracy finite-difference (HAFD) numerical method. We present a strong-stability-preserving finite-difference scheme and compare its performance, using some relevant examples, with those of two recently published numerical methods for solving this kind of equation: a simpler second order finite-difference method and a pseudospectral numerical scheme that enforces the conservation of energy. The numerical comparison shows that the three methods have similar accuracy for short times, but the simpler finite-difference scheme is less accurate in preserving the three invariants, affecting the numerical accuracy of the solution as time goes on. On the contrary, the HAFD method presented here preserves the invariants with even better accuracy than the pseudospectral scheme, but with a much lower computational cost. In addition, the numerical implementation of the HAFD method is as easy as that of the simpler finite-difference method, being both much simpler than the intricate energy conservation pseudospectral scheme. These advantages makes the HAFD method presented here very appropriate for solving numerically this type of equations, particularly for studying long time wave propagation.

Fully pseudospectral solution of the conformally invariant wave equation near the cylinder at spacelike infinity

We study the scalar, conformally invariant wave equation on a four-dimensional Minkowski background in spherical symmetry, using a fully pseudospectral numerical scheme. Thereby, our main interest is in a suitable treatment of spatial infinity, which is represented as a cylinder. We consider both Cauchy problems, where we evolve data from a Cauchy surface to future null infinity, as well as characteristic initial value problems with data at past null infinity, and demonstrate that highly accurate numerical solutions can be obtained for a small number of grid points.

Kardar-Parisi-Zhang asymptotics for the two-dimensional noisy Kuramoto-Sivashinsky equation

We study numerically the Kuramoto-Sivashinsky equation forced by external white noise in two space dimensions, that is a generic model for, e.g., surface kinetic roughening in the presence of morphological instabilities. Large scale simulations using a pseudospectral numerical scheme allow us to retrieve Kardar-Parisi-Zhang (KPZ) scaling as the asymptotic state of the system, as in the one-dimensional (1D) case. However, this is only the case for sufficiently large values of the coupling and/or system size, so that previous conclusions on non-KPZ asymptotics are demonstrated as finite size effects. Crossover effects are comparatively stronger for the two-dimensional case than for the 1D system.

A PSEUDO-SPECTRAL NUMERICAL SCHEME FOR THE SIMULATION OF STEADY AND OSCILLATING WALL-BOUNDED FLOWS

The objective of the present work is to present and validate a pseudo-spectral numerical scheme based on a variational formulation for the solution of three-dimensional, time-dependent wall-bounded forced and natural convective flows. One of the novel aspects of this numerical scheme is the use of rescaled Legendre-Lagrangian interpolants to represent the velocity and temperature in one direction. These interpolants were obtained by dividing the Legendre Lagrangian interpolants of same order by the square root of the corresponding weight used for Gauss-Lobatto quadrature. Results from two specific problems are presented as part of the validation process: Rayleigh-Bénard convection and steady and unsteady channel flow driven by an external oscillating streamwise pressure gradient.

Sedimentation potential of a concentrated spherical colloidal suspension

The sedimentation behavior of a concentrated suspension of charged spherical particles is investigated theoretically. The sedimentation potential is evaluated taking the effect of double layer polarization into account. A cell model is adopted to describe the present multientities system, and a pseudospectral numerical scheme is used to solve the governing electrokinetic equations, which comprise a Navior–Stokes equation for flow field and a Poisson equation for electric field. Several interesting phenomena, which are absent if double layer polarization is neglected, are observed. For example, the ratio (E*/U*), E*, and U* being, respectively, the scaled sedimentation potential and the scale velocity, has a minimum if the electrical potential is low, and has a maximum if it is high.

Plane-Wave Analysis and Comparison of Split-Field, Biaxial, and Uniaxial PML Methods as ABCs for Pseudospectral Electromagnetic Wave Simulations in Curvilinear Coordinates

In this paper, we discuss and compare split-field, biaxial, and uniaxial perfectly matched layer (PML) methods for absorbing outgoing vector waves in cylindrical and spherical coordinates. We first extend Berenger's split-field formulation into spherical and cylindrical coordinates in such a way that it maintains all the desirable properties it exhibits in rectangular coordinates. Then we discuss the biaxial and the uniaxial medium PML methods in Cartesian coordinates and extend them to spherical and cylindrical coordinates. Properties of plane-wave solutions of the PML methods are analyzed. In particular, the decay and boundness properties of the solutions are considered in order to provide further insight into the different formulations presented herein. Moreover, we propose a set of symmetric hyperbolic equations for both the biaxial and the uniaxial PML methods in the time-domain, which is fine-tuned in numerical experiments and very suitable for time-domain problems. All three types of spherical and cylindrical PML methods are applied in simulations of plane wave scattering as well as radiating dipole problems. We use a multidomain pseudospectral (Chebyshev) numerical scheme, and the effectiveness of the PML methods is demonstrated through the accurate numerical results obtained. The order of outer-boundary reflection is as low as 0.1% of the exact solution.

Double-diffusive convection in a rotating annulus with horizontal temperature and vertical solutal gradients

A numerical study is made of double-diffusive convection in a rotating annulus. Motions are driven by the externally applied horizontal temperature gradient. The stable solutal gradient is aligned in the vertical direction. Parametric studies are performed in order to acquire an understanding of the qualitative character of the axisymmetric basic state of the resulting flow. The aim is to inquire as to the effect of rotation on the global flow structure. A high-accuracy pseudospectral numerical scheme is employed to integrate the axisymmetric incompressible Navier-Stokes equations. Computational results are presented to disclose the detailed fields of azimuthal and meridional flows, temperature, and solute. A total of nine parameter sets, produced by a combination of three values of stratification number and three buoyancy ratios, were dealt with in the computation. The Prandtl number was set Pr = 1.0, the thermal Rayleigh number Ra T = 10 5 , and the Lewis number Le = 10.0. For a low buoyancy ratio, motions are vigorous, especially when the rotation effect is small, resembling a sidewall-heated pure thermal convection. In the interior, a linear temperature stratification and a well-mixed solutal field are seen. When the buoyancy ratio is moderate, the overall character of flow shows considerable dependence on the relative strength of rotation effect. For a large buoyancy ratio, the stabilizing solutal effect is dominant. The convective activities are very weak, and heat transfer is mostly conductive.

Steady-state simulation of 2D homogeneous turbulence

In contrast to the pseudo-spectral numerical scheme, we have devised a new way of dealing with the 2D Navier-Stokes equations in wavevector space. Our approach is based on the original spectral formulation, but the reduction of the Navier-Stokes equations to ordinary differential equations is carried out by symbolic manipulations on a computer. This permits us to leave out certain Fourier amplitudes and triad-interactions which may be unimportant for the averaged dynamics. By retaining certain subsets of wavevectors with doubling wavenumbers, abridged equations of motion can be constructed systematically. Preliminary simulations have shown the attainment of k −2 equilibrium energy distribution under no forcing and damping, and the simultaneous development of k − 5 3 and k −3 energy spectra when forced to counteract viscous damping.

Route to chaos in porous-medium thermal convection

A pseudo-spectral numerical scheme is used to study two-dimensional, single-cell, time-dependent convection in a square cross-section of fluid saturated porous material heated from below. With increasing Rayleigh number R convection evolves from steady S to chaotic NP through the sequence of bifurcations S→P(1)→QP2→P(2)→NP, where P(1) and P(2) are simply periodic regimes and QP2 is a quasi-periodic state with two basic frequencies. The transitions (from onset of convection to chaos) occur at Rayleigh numbers of 4π2, 380–400, 500–520, 560–570, and 850–1000. In the first simply periodic regime the fundamental frequency f1 varies as . The chaotic states are characterized by spectral peaks with at least 3 fundamental frequencies superimposed on a broadband background noise. The time dependence of these states arises from the random generation of tongue-like disturbances within the horizontal thermal boundary layers. Transition to the chaotic regime is accompanied by the growth of spectral components that destroy the centre-symmetry of convection in the other states. Over-truncation can lead to spurious transitions and bifurcation sequences; in general it produces overly complex flows.

Spectral Methods in Time for Hyperbolic Equations

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.

A pseudospectral scheme for the numerical calculation of shocks

A pseudospectral numerical scheme is developed in order to calculate the propagation of a shock wave. We use a two-step time-differencing method and a Chebyshev transform method to compute the space derivative. Such a scheme has been used to reduce the oscillations due to Gibb's phenomenon. This numerical method is applied to the solution of the Burgers equation without viscosity term. The accuracy of the numerical solutions is compared to the one given by two finite difference methods.