The modified decomposition method and Padé approximants for the MHD flow over a non-linear stretching sheet

In this work, the magnetohydrodynamic (MHD) boundary layer flow is investigated by employing the modified Adomian decomposition method and the Padé approximation. The series solution of the governing non-linear problem is developed. Comparison of the present solution is made with the existing solution and excellent agreement is noted.     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.     

Multi-level spectral Galerkin method for the Navier–Stokes equations, II: time discretization

A fully discrete multi-level spectral Galerkin method in space–time for the two-dimensional nonstationary Navier–Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier–Stokes problem is only solved on the lowest-dimensional space with the largest time step Δt ₁; subsequent approximations are generated on a succession of higher-dimensional spaces with small time step Δt _j by solving a linearized Navier–Stokes problem about the solution on the previous level. Some error estimates are also presented for the J-level spectral Galerkin method. The scaling relations of the dimensional numbers and time mesh widths that lead to optimal accuracy of the approximate solution in H ¹-norm and L ²-norm are investigated, i.e., m _j∼m _j−1 ^3/2 , Δt _j∼Δt _j−1 ^3/2 , j=2,. . .,J. We demonstrate theoretically that a fully discrete J-level spectral Galerkin method is significantly more efficient than the standard one-level spectral Galerkin method. Mathematics subject classifications (2000) 35L70, 65N30, 76D06 Subsidized by the Special Funds for Major State Basic Research Projects G1999032801-07, NSF of China 10371095 and the City University of Hong Kong Research Project 7001093, NSF of China 50323001.     

A generalized Laguerre–Legendre spectral collocation method for solving initial-boundary value problems

In this research, a mixed spectral collocation method based on Kronecker product is proposed for solving initial-boundary value problems. New implementation is suggested to achieve more accurate approximation at longer times. Test problems are also studied to demonstrate how this method is implemented. Numerical experiments reveal that the new method is very effective and convenient.     

On the stagnation-point flow towards a stretching sheet with homogeneous–heterogeneous reactions effects

The effects of homogeneous–heterogeneous reactions on the steady boundary layer flow near the stagnation point on a stretching surface is studied. The possible steady-states of this system are analyzed in the case when the diffusion coefficients of both reactant and auto catalyst are equal. The strength of this effect is represented by the dimensionless parameter K and K_s. It is shown that for a fluid of small kinematic viscosity, a boundary layer is formed when the stretching velocity is less than the free stream velocity and an inverted boundary layer is formed when the stretching velocity exceeds the free stream velocity. The uniqueness of this problem lies on the fact that the solutions are possible for all values of λ > 0 (stretching surface), while for λ < 0 (shrinking surface), solutions are possible only for its limited range.     

Comment on “Existence and uniqueness results for a nonlinear differential equation arising in viscous flow over a nonlinearly stretching sheet”

The article named above appeared recently in Applied Mathematics Letters and investigated a boundary value problem governing viscous flow over a nonlinearly stretching sheet. The authors of the work assert existence and (under certain restrictions) uniqueness of a solution to the problem for all relevant values of the parameter governing the stretching rate of the sheet. Unfortunately, several proofs presented in the article are incorrect. We will prove that for a range of parameter space the solution to the BVP is not unique. For these parameter values there are infinitely many solutions to the problem. The same incorrect analysis is reproduced in several other papers (see the references). Some of the claims of these papers are contradicted by established results on, for example, the Falkner–Skan problem.     

Mass and momentum conservation of the least-squares spectral collocation method for the Navier–Stokes equations

From the literature, it is known that the Least-Squares Spectral Element Method (LSSEM) for the stationary Stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum. Furthermore, it is known that the Least-Squares Spectral Collocation Method (LSSCM) leads to superior conservation of mass and momentum for the stationary Stokes equations. In the present paper, we consider mass and momentum conservation of the LSSCM for time-dependent Stokes and Navier–Stokes equations. We observe that the LSSCM leads to improved conservation of mass (and momentum) for these problems. Furthermore, the LSSCM leads to the well-known time-dependent profiles for the velocity and the pressure profiles. To obtain these results, we use only a few elements, each with high polynomial degree, avoid normal equations for solving the overdetermined linear systems of equations and introduce the Clenshaw–Curtis quadrature rule for imposing the average pressure to be zero. Furthermore, we combined the transformation of Gordon and Hall (transfinite mapping) with the least-squares spectral collocation scheme to discretize the internal flow problems.     

A Legendre spectral element method on a large spatial domain to solve the predator–prey system modeling interacting populations

A Legendre spectral element method is developed for solving a one-dimensional predator–prey system on a large spatial domain. The predator–prey system is numerically solved where the prey population growth is described by a cubic polynomial and the predator’s functional response is Holling type I. The discretization error generated from this method is compared with the error obtained from the Legendre pseudospectral and finite element methods. The Legendre spectral element method is also presented where the predator response is Holling type II and the initial data are discontinuous.     

The least squares spectral element method for the Cahn–Hilliard equation

The problem of numerically resolving an interface separating two different components is a common problem in several scientific and engineering applications. One alternative is to use phase field or diffuse interface methods such as the Cahn–Hilliard (C–H) equation, which introduce a continuous transition region between the two bulk phases. Different numerical schemes to solve the C–H equation have been suggested in the literature. In this work, the least squares spectral element method (LS-SEM) is used to solve the Cahn–Hilliard equation. The LS-SEM is combined with a time–space coupled formulation and a high order continuity approximation by employing C¹¹p-version hierarchical interpolation functions both in space and time. A one-dimensional case of the Cahn–Hilliard equation is solved and the convergence properties of the presented method analyzed. The obtained solution is in accordance with previous results from the literature and the basic properties of the C–H equation (i.e. mass conservation and energy dissipation) are maintained. By using the LS-SEM, a symmetric positive definite problem is always obtained, making it possible to use highly efficient solvers for this kind of problems. The use of dynamic adjustment of number of elements and order of approximation gives the possibility of a dynamic meshing procedure for a better resolution in the areas close to interfaces.     

An Alternating-Direction Sinc–Galerkin method for elliptic problems

A new Alternating-Direction Sinc–Galerkin (ADSG) method is developed and contrasted with classical Sinc–Galerkin methods. It is derived from an iterative scheme for solving the Lyapunov equation that arises when a symmetric Sinc–Galerkin method is used to approximate the solution of elliptic partial differential equations. We include parameter choices (derived from numerical experiments) that simplify existing alternating-direction algorithms. We compare the new scheme to a standard method employing Gaussian elimination on a system produced using the Kronecker product and Kronecker sum, as well as to a more efficient algorithm employing matrix diagonalization. We note that the ADSG method easily outperforms Gaussian elimination on the Kronecker sum and, while competitive with matrix diagonalization, does not require the computation of eigenvalues and eigenvectors.     

A modified Lagrange–Galerkin method for a fluid-rigid system with discontinuous density

In this paper, we propose a new characteristics method for the discretization of the two dimensional fluid-rigid body problem in the case where the densities of the fluid and the solid are different. The method is based on a global weak formulation involving only terms defined on the whole fluid-rigid domain. To take into account the material derivative, we construct a special characteristic function which maps the approximate rigid body at the (k?+?1)-th discrete time level into the approximate rigid body at k-th time. Convergence results are proved for both semi-discrete and fully-discrete schemes.     

Hybridized schemes of the discontinuous Galerkin method for stationary convection–diffusion problems

For stationary linear convection–diffusion problems, we construct and study a new hybridized scheme of the discontinuous Galerkin method on the basis of an extended mixed statement of the problem. Discrete schemes can be used for the solution of equations degenerating in the leading part and are stated via approximations to the solution of the problem, its gradient, the flow, and the restriction of the solution to the boundaries of elements. For the spaces of finite elements, we represent minimal conditions responsible for the solvability, stability, accuracy, and superconvergence of the schemes. A new procedure for the post-processing of solutions of HDG-schemes is suggested.     

A Petrov–Galerkin RBF method for diffusion equation on the unit sphere

This paper concerns a numerical solution for the diffusion equation on the unit sphere. The given method is based on the spherical basis function approximation and the Petrov–Galerkin test discretization. The method is meshless because spherical triangulation is not required neither for approximation nor for numerical integration. This feature is achieved through the spherical basis function approximation and the use of local weak forms instead of a global variational formulation. The local Petrov–Galerkin formulation allows to compute the integrals on small independent spherical caps without any dependence on a connected background mesh. Experimental results show the accuracy and the efficiency of the new method.     

Perfectly matched layers for the heat and advection–diffusion equations

We design a perfectly matched layer for the advection–diffusion equation. We show that the reflection coefficient is exponentially small with respect to the damping parameter and the width of the PML and independently of the advection and of the viscosity parameters. Numerical tests assess the efficiency of the approach.