A Jacobi–Davidson method for two‐real‐parameter nonlinear eigenvalue problems arising from delay‐differential equations

The critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.     

The Sinc‐Galerkin method for solving an inverse parabolic problem with unknown source term

The inverse problem of determining an unknown source term depending on space variable in a parabolic equation is considered. A numerical algorithm is presented for recovering the unknown function and obtaining a solution of the problem. As this inverse problem is ill‐posed, Tikhonov regularization is used for finding a stable solution. For solving the direct problem, a Galerkin method with the Sinc basis functions in both the space and time domains is presented. This approximate solution displays an exponential convergence rate and is valid on the infinite time interval. Finally, some examples are presented to illustrate the ability and efficiency of this numerical method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

The main purpose of this work is to investigate an initial boundary value problem related to a suitable class of variable order fractional integro‐partial differential equations with a weakly singular kernel. To discretize the problem in the time direction, a finite difference method will be used. Then, the Sinc‐collocation approach combined with the double exponential transformation is employed to solve the problem in each time level. The proposed numerical algorithm is completely described and the convergence analysis of the numerical solution is presented. Finally, some illustrative examples are given to demonstrate the pertinent features of the proposed algorithm.     

A numerical study of the semi‐classical limit for three‐coupled long wave–short wave interaction equations

We study numerically the semi‐classical limit for three‐coupled long wave–short wave interaction equations. The Fourier–Galerkin semi‐discretization is proved to be spectrally convergent in an appropriate energy space. We propose a split‐step Fourier method in the semi‐classical regime with the discussion of the meshing strategy, which is necessary to obtain correct numerical solution. Plane wave solution with weak and strong initial phases, solitary wave solution and Gaussian solution are considered to investigate the semi‐classical limit.     

Recurrent solutions of the linearly coupled complex cubic‐quintic Ginzburg‐Landau equations

In this paper, we will establish the bounded solutions, periodic solutions, quasiperiodic solutions, almost periodic solutions, and almost automorphic solutions for linearly coupled complex cubic‐quintic Ginzburg‐Landau equations, under suitable conditions. The main difficulty is the nonlinear terms in the equations that are not Lipschitz‐continuity, traditional methods cannot deal with the difficulty in our problem. We overcome this difficulty by the Galerkin approach, energy estimate method, and refined inequality technique.     

Bernstein Ritz‐Galerkin method for solving an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

In this article, the Ritz‐Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then Ritz‐Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

The advection‐diffusion equation has a long history as a benchmark for numerical methods. Taylor‐Galerkin methods are used together with the type of splines known as B‐splines to construct the approximation functions over the finite elements for the solution of time‐dependent advection‐diffusion problems. If advection dominates over diffusion, the numerical solution is difficult especially if boundary layers are to be resolved. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method with emphasis on treatment of boundary conditions. Taylor‐Galerkin methods have been constructed by using both linear and quadratic B‐spline shape functions. Results shown by the method are found to be in good agreement with the exact solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A uniformly optimal‐order error estimate for a bilinear finite element method for degenerate convection‐diffusion equations

We prove an optimal‐order error estimate in a degenerate‐diffusion weighted energy norm for bilinear Galerkin finite element methods for two‐dimensional time‐dependent convection‐diffusion equations with degenerate diffusion. In the estimate, the generic constants depend only on certain Sobolev norms of the true solution but not the lower bound of the diffusion. This estimate, combined with a known stability estimate of the true solution of the governing partial differential equations, yields an optimal‐order estimate of the Galerkin finite element method, in which the generic constants depend only on the Sobolev norms of the initial and right side data. Preliminary numerical experiments were conducted to verify these estimates numerically. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

Stability and convergence of optimum spectral non‐linear Galerkin methods

Our objective in this article is to present some numerical schemes for the approximation of the 2‐D Navier–Stokes equations with periodic boundary conditions, and to study the stability and convergence of the schemes. Spatial discretization can be performed by either the spectral Galerkin method or the optimum spectral non‐linear Galerkin method; time discretization is done by the Euler scheme and a two‐step scheme. Our results show that under the same convergence rate the optimum spectral non‐linear Galerkin method is superior to the usual Galerkin methods. Finally, numerical example is provided and supports our results. Copyright © 2001 John Wiley & Sons, Ltd.     

On order‐reducible sinc discretizations and block‐diagonal preconditioning methods for linear third‐order ordinary differential equations

By introducing a variable substitution, we transform the two‐point boundary value problem of a third‐order ordinary differential equation into a system of two second‐order ordinary differential equations (ODEs). We discretize this order‐reduced system of ODEs by both sinc‐collocation and sinc‐Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs. The coefficient matrix of the linear system is of block two‐by‐two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block‐diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.     

A CFL‐free explicit characteristic interior penalty scheme for linear advection‐reaction equations

We develop a CFL‐free, explicit characteristic interior penalty scheme (CHIPS) for one‐dimensional first‐order advection‐reaction equations by combining a Eulerian‐Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal‐order error estimate in the L² norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Combination of standard Galerkin and subspace methods for the time‐dependent Navier‐Stokes equations with nonsmooth initial data

We consider a combination of the standard Galerkin method and the subspace decomposition methods for the numerical solution of the two‐dimensional time‐dependent incompressible Navier‐Stokes equations with nonsmooth initial data. Because of the poor smoothness of the solution near t = 0, we use the standard Galerkin method for time interval [0, 1] and the subspace decomposition method time interval [1, ∞). The subspace decomposition method is based on the solution into the sum of a low frequency component integrated using a small time step Δt and a high frequency integrated using a larger time step pΔt with p > 1. From the H¹‐stability and L²‐error analysis, we show that the subspace decomposition method can yield a significant gain in computing time. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

An Eulerian‐Lagrangian discontinuous Galerkin method for transient advection‐diffusion equations

We develop an Eulerian‐Lagrangian discontinuous Galerkin method for time‐dependent advection‐diffusion equations. The derived scheme has combined advantages of Eulerian‐Lagrangian methods and discontinuous Galerkin methods. The scheme does not contain any undetermined problem‐dependent parameter. An optimal‐order error estimate and superconvergence estimate is derived. Numerical experiments are presented, which verify the theoretical estimates.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

Time‐dependent algorithms for viscoelastic flow: Finite element/volume schemes

Hybrid finite volume/element methods are investigated within the context of transient viscoelastic flows. A finite volume algorithm is proposed for the hyperbolic constitutive equation, of Oldroyd‐form, whereas the continuity/momentum balance is accommodated through a Taylor‐Galerkin finite element method. Various finite volume combinations are considered to derive accurate and stable implementations. Consistency of formulation is key, embracing fluctuation distribution and median‐dual‐cell constructs, within a cell‐vertex discretisation on triangles. In addition, we investigate the effect of treating the time‐term in a finite element fashion, using mass‐matrix iteration instead of the standard finite volume mass‐lumping approach. We devise an accurate transient scheme that captures the analytical solution at short and long time, both in core flow and near shear boundaries. In this respect, some difficulties are highlighted. A new method emerges, with the Low Diffusion B (LDB, with or without mass‐matrix iteration) as the optimal choice. We progress to a complex flow application and demonstrate some provocative features due to the influence of true transient boundary conditions on evolutionary flow‐structure in a 4:1 start‐up rounded‐corner contraction problem. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A non‐commutative Landau‐Zener formula

We study semi‐classical measures of families of solutions to a 2 × 2 Dirac system with 0 mass, which presents bands crossing. We focus on constant electro‐magnetic fields. The fact that these fields are orthogonal or not leads to different geometric situations. In the first case, one reduces to some well‐understood model problem. For studying the second case, we introduce some two‐scale semi‐classical measures associated with symplectic submanifold. These measures are operator‐valued measures and the transfer of energy at the crossing is described by a non‐commutative Landau‐Zener formula for these measures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second‐order Galerkin‐Lagrange method for the Navier‐Stokes equations (retracted article)

It has come to the attention of the editors and publisher that an article published in Numerical Methods and Partial Differential Equations, “Second‐order Galerkin‐Lagrange method for the Navier‐Stokes equations,” by Mohamed Bensaada, Driss Esselaoui, and Pierre Saramito, Numer Methods Partial Differential Eq 21(6) (2005), 1099–1121 included large portions that were copied from the following paper without proper citation: “Convergence and nonlinear stability of the Lagrange‐Galerkin method for the Navier‐Stokes equations,” Endre Suli, Numerische Mathematik, Vol. 53, No. 4, pp. 459–486 (July, 1988). We have retracted the paper and apologize to Dr. Suli Numer Methods Partial Differential Eq (2007)23(1)211 .     

Well‐posedness of the equations governing the motions of a one‐dimensional hybrid thermo‐elastic structure

In this paper a model for the vibrations of a one‐dimensional hybrid thermo‐elastic structure consisting of an extensible thermo‐elastic beam which is hinged at one end, with a rigid body attached to its free end, is studied with a view to establishing the existence of a unique solution in a weak sense. The model takes account of the effect of stretching on bending and rotational inertia. By treating eigenvalue problems with the spectral parameter also in the boundary conditions, we are able to employ the method of Faedo–Galerkin approximations. Copyright © 2004 John Wiley & Sons, Ltd.     

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

We propose a nonintrusive reduced‐order modeling method based on the notion of space‐time‐parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time‐dependent partial differential equations. A two‐level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced‐order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced‐order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection‐reaction‐diffusion problem. We demonstrate that our approach leads to reduced‐order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Numerical simulations for two‐dimensional incompressible Navier‐Stokes equations with stochastic boundary conditions

The article mainly concerns modeling the stochastic input and its propagation in incompressible Navier‐Stokes(N‐S) flow simulations. The stochastic input is represented spectrally by employing orthogonal polynomial functionals from the Askey scheme as trial basis to represent the random space. A standard Galerkin projection is applied in the random dimension to derive the equations in the weak form. The resulting set of deterministic equations is then solved with standard methods to obtain the mean solution and variance of the stochastic velocity. In this article, the main method employs the Hermite polynomial as the basis in random space. Cavity problems are given to demonstrate the process of numerical simulation. Furthermore, Monte‐Carlo simulation method is applied to illustrate the accurate numerical results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Determination of an unknown function in a parabolic inverse problem by Sinc‐collocation method

In this article, an inverse problem of determining an unknown time‐dependent source term of a parabolic equation is considered. We change the inverse problem to a Volterra integral equation of convolution‐type. By using Sinc‐collocation method, the resulting integral equation is replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the condition number and the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. Some examples are given to demonstrate the computational efficiency of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1584–1598, 2010