Multidomain Spectral Collocation Method Research Articles

A multi-domain spectral collocation method for the Fokker–Planck equation in an infinite channel

In this paper, we propose a multi-domain spectral collocation method for partial differential equations on two-dimensional unbounded domains. Some approximation results on the composite generalized Laguerre-Legendre interpolation and quasi-orthogonal projecting are established, respectively. These results play a significant role in related spectral collocation method. As an application, a multi-domain spectral collocation scheme is provided for the Fokker–Planck equation with absorption or non-homogeneous boundary conditions. The convergence of the proposed algorithm is performed. An efficient implementation is presented. Numerical experiments demonstrate the effectiveness and high accuracy of the algorithm.

A numerical study of heat and mass transfer in a Darcy porous medium saturated with a couple stress fluid under rotational modulation

A theoretical investigation of the influence of time-dependent rotational modulation on the stability, heat, and mass transfer in a Darcy porous medium with a couple stress fluid is presented. A perturbation technique dependent on the infinitesimal amplitude of the disturbance parameter is used to study the consolidated effects of modulating rotation, solute gradient, and permeability on the stability of a porous medium. The Ginzburg Landau equation is obtained and used to quantify heat and mass transport as the Nusselt and Sherwood numbers respectively. The coupled nonlinear Lorenz equations are solved using a newly developed multidomain spectral collocation method. Simulations of the phase plane trajectories are used to represent and analyze limited sets of solutions. We find among other results, that the effect of increasing the modulation amplitude is to accelerate the heat and mass transport. The results from the Ginzburg Landau approach are qualitatively similar to that of the Lorenz approach although the Lorenz approach was more efficient.

On trigonometric cosine, square, sawtooth, and triangular wave‐type rotational modulations on triple‐diffusive convection in salted water

AbstractThe effect of sinusoidal (trigonometric cosine) and nonsinusoidal (square, sawtooth, and triangular) rotational modulations on triple‐diffusive convection in a Newtonian fluid is studied in this paper. The derived Ginzburg–Landau equation for the stationary mode of convection is used to quantify heat and mass transport. The enhancement of heat and mass transport in water due to the addition of different salts is explained in terms of the thermophysical properties of the fluid and the aqueous solutes. The flow equations are reduced to a Lorenz‐type scheme of nonlinear evolution equations using a truncated Fourier series. The equations describing the growth of convection amplitudes are solved using the multidomain spectral collocation method. With the aid of thermophysical properties of the fluid, numerical computations are performed for different values of the revised Rayleigh number (), Taylor number (), the amplitude of modulation , and modulation frequency . A comparison of heat and mass transport in different combinations of aqueous solutions is made. The study reveals that all modes of rotational modulation can control the rate of heat and mass transport. It is noted that amongst all other modulations, the square wave modulation is the most destabilizing one.

A multi-domain spectral collocation method for Volterra integral equations with a weakly singular kernel

In this paper, we introduce a multi-domain Müntz-polynomial spectral collocation method with graded meshes for solving second kind Volterra integral equations with a weakly singular kernel. This method is particularly suitable for problems whose solutions contain non-integer exponent factors. We carry out a rigorous error analysis of hp-version in the L∞- and weighted L2-norms. Several numerical examples are presented to demonstrate the efficiency and accuracy of the method.

Multi-domain spectral collocation method for variable-order nonlinear fractional differential equations

Spectral and spectral element methods using Galerkin type formulations are efficient for solving linear fractional PDEs (FPDEs) of constant order but are not efficient in solving nonlinear FPDEs and cannot handle FPDEs with variable-order. In this paper, we present a multi-domain spectral collocation method that addresses these limitations. We consider FPDEs in the Riemann–Liouville sense, and employ Jacobi Lagrangian interpolants to represent the solution in each element. We provide variable-order differentiation formulas, which can be computed efficiently for the multi-domain discretization taking into account the nonlocal interactions. We enforce the interface continuity conditions by matching the solution values at the element boundaries via the Lagrangian interpolants, and in addition we minimize the jump in (integer) fluxes using a penalty method. We analyze numerically the effect of the penalty parameter on the condition number of the global differentiation matrix and on the stability and convergence of the penalty collocation scheme. We demonstrate the effectiveness of the new method for the fractional Helmholtz equation of constant and variable-order using h−p refinement for different values of the penalty parameter. We also solve the fractional Burgers equation with constant and variable-order and compare with solutions obtained with a single domain spectral collocation method.

Thermo-convective instability in a rotating ferromagnetic fluid layer with temperature modulation

Abstract We study the thermoconvective instability in a rotating ferromagnetic fluid confined between two parallel infinite plates with temperature modulation at the boundaries. We use weakly nonlinear stability theory to analyze the stationary convection in terms of critical Rayleigh numbers. The influence of parameters such as the Taylor number, the ratio of the magnetic force to the buoyancy force and the magnetization on the flow behaviour and structure are investigated. The heat transfer coefficient is analyzed for both the in-phase and the out-of-phase modulations. A truncated Fourier series is used to obtain a set of ordinary differential equations for the time evolution of the amplitude of convection for the ferromagnetic fluid flow. The system of differential equations is solved using a recent multi-domain spectral collocation method that has not been fully tested on such systems before. The solutions sets are presented as sets of trajectories in the phase plane. For some supercritical values of the Rayleigh number, spiralling trajectories that transition to chaotic solutions are obtained. Additional results are presented in terms of streamlines and isotherms for various Rayleigh numbers.

An accelerated Poisson solver based on multidomain spectral discretization

This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The method works best for domains that can readily be mapped onto a rectangle, or a collection of nonoverlapping rectangles. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and higher have been tested and work well). Local mesh refinement results in highly accurate solutions even in the presence of local irregular behavior due to corner singularities, localized loads, etc. The system of linear equations attained upon discretization is solved using a direct (as opposed to iterative) solver with O(N^{1.5}) complexity for the factorization stage and O(N log N) complexity for the solve. The scheme is ideally suited for executing the elliptic solve required when parabolic problems are discretized via time-implicit techniques. In situations where the geometry remains unchanged between time-steps, very fast execution speeds are obtained since the solution operator for each implicit solve can be pre-computed.

Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method

A full-vectorial mode solver in terms of the transverse magnetic field components for optical waveguides with transverse anisotropy is described by using the multidomain spectral collocation method based on Chebyshev polynomials. The waveguide cross section surrounded by the perfectly matched layers is divided into suitable number of homogeneous rectangles, and then connected with by imposing the continuities of the longitudinal field components at the dielectric interfaces shared by the adjacent rectangles, resulting in a generalized matrix eigenvalue problem. To validate the established method, results of an anisotropic square waveguide and a magnetooptic rib waveguide are presented and compared with those from the full-vectorial finite difference method, full-vectorial beam propagation method, and the experimental data.

A mass conserving multi-domain spectral collocation method for the Stokes problem

A muli-domain spectral collocation method is developed for the approximation of the Stokes problem in two dimensions. Compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well-posed. The collocation scheme possesses the property that the continuity equation is satisfied at all the collocation points. In addition, for problems defined in domains which are rectangularly decomposable, the scheme yields velocity approximations that are globally divergence-free.

A Conservative Staggered-Grid Chebyshev Multidomain Method for Compressible Flows

We present a new multidomain spectral collocation method that uses a staggered grid for the solution of compressible flow problems. The solution unknowns are defined at the nodes of a Gauss quadrature rule. The fluxes are evaluated at the nodes of a Gauss–Lobatto rule. The method is conservative, free-stream preserving, and exponentially accurate. A significant advantage of the method is that subdomain corners are not included in the approximation, making solutions in complex geometries easier to compute.

An implicit multidomain spectral collocation method for stiff highly non-linear fluid dynamics problems

An implicit, multidomain, Chebyshev spectral collocation technique for solving highly non-linear stiff problems is presented. First, the accuracy and efficiency of the explicit—implicit Adams—Bashforth Crank—Nicolson (ABCN) and the implicit Crank—Nicolson (CN) methods in obtaining solutions to the Burgers equation are compared. Then, the implicit CN scheme is used to solve the highly non-linear Reynolds equation of lubrication for a gas bearing separating two textured surfaces. Surface texture is approximated by fifty sinusoidal waves each of which is assigned a separate subdomain. An additional subdomain is used to resolve the trailing edge boundary layer. The combined method is shown to be well suited for the simulation of highly non-linear phenomena that exhibit high frequency spatial variations in the desired solution and that possess very sharp interior or boundary layers.

Multidomain Spectral Solution of Compressible Viscous Flows

We develop a nonoverlapping multidomain spectral collocation method to solve compressible viscous flows. At the interfaces, the advection terms are treated with a characteristic correction method. The diffusion terms are treated with a penalty method. Spectral accuracy is demonstrated on linear model problems in one and two space dimensions. The method is applied to a subsonic and a supersonic flow over a flat plate. The results are compared to solutions of the boundary-layer equations which show that two digit accuracy in the adiabatic plate temperature is obtained with 16 points in the boundary layer for a freestream Mach number of two. A second application is to a transonic flow in a two-dimensional converging-diverging nozzle, where the computed results are compared to experimental data.

Multidomain spectral solutions of high-speed flows over blunt cones

A shock-fitted multidomain spectral collocation method is used to solve steady inviscid supersonic flows over two blunt, axisymmetric cones. The calculations are compared to a conical flow solution and to experimental data

Spectral solution of inviscid supersonic flows over wedges and axisymmetric cones

We use a shock-fitted multidomain spectral collocation method to solve both steady and unsteady inviscid supersonic flows over bodies. New aspects of the method include two subdomain interface types and a zonal solution procedure to get efficient convergence to steady-state in supersonic regions. Steady-state examples include flow over a sharp cone, a hyperbolic cone and a hyperbolic wedge. For the cone, the exact flow solution is used to show that the method is spectrally accurate. As an example of an unsteady flow, we present a calculation of the interaction of a free-stream hot ring with the flow over a sharp cone.

A multidomain spectral collocation method for the Stokes problem

We propose a multidomain spectral collocation scheme for the approximation of the two-dimensional Stokes problem. We show that the discrete velocity vector field is exactly divergence-free and we prove error estimates both for the velocity and the pressure.

Numerical methods for hypersonic boundary layer stability

Various numerical methods for the solution of linear stability equations for compressible boundary layers are compared. Both the global and local eigenvalue methods for temporal stability analysis are discussed. Global methods are used to compute all the eigenvalues of the discretized system. When a guess for the desired eigenvalue is available, local methods may be used both to purify the eigenvalue and compute the associated eigenfunctions. The extension to spatial stability analysis is also considered. The discretizations studied include: a second-order finite-difference method, a fourth-order accurate two-point compact difference scheme, and a Chebyshev spectral collocation method. Eigenvalue results are presented for Mach numbers up to 10. As the Mach number increases, the performance of the spectral method deteriorates due to the outward movement of the critical layer. To alleviate this problem, a multi-domain spectral collocation method is developed which exhibits better convergence. The overall performance of the fourth-order compact scheme is excellent. Our results also indicate that, in the limit of vanishing Mach number, there exist stable discrete modes in addition to the discrete modes of the Orr-Sommerfeld equation.