Soliton perturbation theory for a higher order Hirota equation

Solitary wave evolution for a higher order Hirota equation is examined. For the higher order Hirota equation resonance between the solitary waves and linear radiation causes radiation loss. Soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and compared to numerical solutions. An excellent comparison between numerical and theoretical solutions is obtained for both right- and left-moving waves. Also, a two-parameter family of higher order asymptotic embedded solitons is identified.     

A higher order numerical scheme for scalar transport

The computational principles of a numerical scheme for the solution of the two-dimensional scalar transport equation are presented. The scheme is designed for use in transient flow situations where accurate simulation of the advective process is important. Advective transport is computed by the method of characteristics in which the scalar field is represented by a Hermitian polynomial complete through the third degree in both coordinate directions, while diffusion is computed by central differencing. The superior accuracy of the new method is demonstrated by analysing its propagation characteristics and by comparing its performance on standard test problems with that of some well-known lower order methods. Finally, the method's applicability is demonstrated in several examples involving tracer releases into channel flows. Where possible the results of these simulations are compared with analytical solutions.     

The spectral accuracy of a fully-discrete scheme for a nonlinear third order equation

A time-discrete pseudospectral algorithm is suggested for the numerical solution of a nonlinear third order equation arising in fluidization. The nonlinear stability and convergence of the new scheme are analyzed. Numerical comparisons with available finite-difference methods are also reported which clearly indicate the superiority of the new scheme.     

Solitary wave solutions for a higher order nonlinear Schrödinger equation

We consider a higher order nonlinear Schrödinger equation with third- and fourth-order dispersions, cubic–quintic nonlinearities, self steepening, and self-frequency shift effects. This model governs the propagation of femtosecond light pulses in optical fibers. In this paper, we investigate general analytic solitary wave solutions and derive explicit bright and dark solitons for the considered model. The derived analytical dark and bright wave solutions are expressed in terms of the model coefficients. These exact solutions are useful to understand the mechanism of the complicated nonlinear physical phenomena which are related to wave propagation in a higher-order nonlinear and dispersive Schrödinger system.     

A third order numerical scheme for the two-dimensional sine-Gordon equation

A rational approximant of third order, which is applied to a three-time level recurrence relation, is used to transform the two-dimensional sine-Gordon (SG) equation into a second-order initial-value problem. The resulting nonlinear finite-difference scheme, which is analyzed for stability, is solved by an appropriate predictor–corrector (P–C) scheme, in which the predictor is an explicit one of second order. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behavior of the proposed P–C/MPC schemes is tested numerically on the line and ring solitons known from the bibliography, regarding SG equation and conclusions for both the mentioned schemes regarding the undamped and the damped problem are derived.     

A fourth order numerical scheme for the one-dimensional sine-Gordon equation

A numerical scheme arising from the use of a fourth order rational approximants to the matrix-exponential term in a three-time level recurrence relation is proposed for the numerical solution of the one-dimensional sine-Gordon (SG) equation already known from the bibliography. The method for its implementation uses a predictor–corrector scheme in which the corrector is accelerated by using the already evaluated corrected values modified predictor–corrector scheme. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. Both the predictor and the corrector schemes are analysed for stability. The predictor–corrector/modified predictor–corrector (P-C/MPC) schemes are tested on single and soliton doublets as well as on the collision of breathers and a comparison of the numerical results with the corresponding ones in the bibliography is made. Finally, conclusions for the behaviour of the introduced MPC over the standard P-C scheme are derived.     

Convergence of the finite-element scheme for the equation of internal waves

The Dirichlet boundary-value problem for the equation of internal waves is considered. To solve it, the finite-element method for both space and time variables is selected. This allows obtaining a highly accurate solution. The accuracy of the method is estimated making certain assumptions on the smoothness of the solutions to the differential problem. If piecewise-cubic finite elements are used, the order of accuracy equals three.     

Efficient exponential compact higher order difference scheme for convection dominated problems

Present work is the development of a finite difference scheme based on Richardson extrapolation technique. It gives an exponential compact higher order scheme (ECHOS) for two-dimensional linear convection-diffusion equations (CDE). It uses a compact nine point stencil, over which the governing equations are discretized for both fine and coarse grids. The resulting algebraic systems are solved using a line iterative approach with alternate direction implicit (ADI) procedure. Combining the solutions over fine and coarse grids, initially a sixth order solution over coarse grid points is obtained. The resultant solution is then extended to finer grid by interpolation derived from the difference operator. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be monotone. The higher order accuracy and better rate of convergence of the developed algorithm have been demonstrated by solving numerous model problems.     

Stability radii of higher order positive difference systems

In this paper we study stability radii of positive polynomial matrices under affine perturbations of the coefficient matrices. It is shown that the real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by some examples.     

Solvability of nonlocal boundary value problems for ordinary differential equation of higher order with a -Laplacian

Nonlocal boundary value problems at resonance for a higher order nonlinear differential equation with a p-Laplacian are considered in this paper. By using a new continuation theorem, some existence results are obtained for such boundary value problems. An explicit example is also given in this paper to illustrate the main results.     

Note on nonstability of the linear functional equation of higher order

We provide a complete solution of the problem of Hyers-Ulam stability for a large class of higher order linear functional equations in single variable, with constant coefficients. We obtain this by showing that such an equation is nonstable in the case where at least one of the roots of the characteristic equation is of module 1. Our results are related to the notions of shadowing (in dynamical systems and computer science) and controlled chaos. They also correspond to some earlier results on approximate solutions of functional equations in single variable.     

A finite-element series-expansion technique for linear surface water waves

A numerical method is presented to deal with the propagation of surface water waves in the framework of the linear theory for an inviscid fluid. For particular geometrical configurations of the region in which wave propagation occurs, refraction, diffraction and reflection phenomena can arise simultaneously, so that the solution of the original Berkhoff equation with appropriate boundary conditions becomes essential to achieve an adequate picture of the resulting field. The method is based on a finite element scheme, in which the element matrices are computed by a series expansion technique. The elements are of arbitrary shape, although of constant depth, and two independent numerical approximations are given for the surface-elevation and velocity fields. An application of the method to the propagation of short water waves in a channel connecting two basins of larger dimensions shows that the method can deal with very large domains, at least when compared to the possibilities of the usual finite element approaches.     

Convergence analysis and numerical implementation of a second order numerical scheme for the three-dimensional phase field crystal equation

In this paper we analyze and implement a second-order-in-time numerical scheme for the three-dimensional phase field crystal (PFC) equation. The numerical scheme was proposed in Hu et al. (2009), with the unique solvability and unconditional energy stability established. However, its convergence analysis remains open. We present a detailed convergence analysis in this article, in which the maximum norm estimate of the numerical solution over grid points plays an essential role. Moreover, we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain, and various three-dimensional numerical results are presented, including the numerical convergence test, complexity test of the multigrid solver and the polycrystal growth simulation.     

Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations

In the context of normalized variable formulation (NVF) of Leonard and total variation diminishing (TVD) constraints of Harten, this paper presents an extension of a previous work by the authors for solving unsteady incompressible flow problems. The main contributions of the paper are threefold. First, it presents the results of the development and implementation of a bounded high order upwind adaptative QUICKEST scheme in the 3D robust code (Freeflow), for the numerical solution of the full incompressible Navier–Stokes equations. Second, it reports numerical simulation results for 1D shock tube problem, 2D impinging jet and 2D/3D broken dam flows. Furthermore, these results are compared with existing analytical and experimental data. And third, it presents the application of the numerical method for solving 3D free surface flow problems.     

Stability of the totally implicit finite difference scheme for the one-dimensional diffusion equation with a moving boundary

We analyze the totally implicitO(h ² k) finite difference method applied to a linear unidimensional diffusion equation with a boundary moving with constant velocity. The related physical problem is the solidification of a finite column filled with a binary mixture, in the quasiequilibrium regime. By advancing the boundary along the pathh/k=velocity of the interface, a simple algorithm is obtained which is shown to be consistent and unconditionally stable for any value of the interfacial segregation factor and of the Peclet number.     

Stability properties of second order delay integro-differential equations

A basic theorem on the behavior of solutions of scalar linear second order delay integro-differential equations is established. As a consequence of this theorem, a stability criterion is obtained.     

Null controllability for a fourth order parabolic equation

In the paper, the null interior controllability for a fourth order parabolic equation is obtained. The method is based on Lebeau-Rabbiano inequality which is a quantitative unique continuation property for the sum of eigenfunctions of the Laplacian.     

Lagrange–Galerkin method for unsteady free surface water waves

We present a new numerical technique to approximate solutions to unsteady free surface flows modelled by the two-dimensional shallow water equations. The method we propose in this paper consists of an Eulerian–Lagrangian splitting of the equations along the characteristic curves. The Lagrangian stage of the splitting is treated by a non-oscillatory modified method of characteristics, while the Eulerian stage is approximated by an implicit time integration scheme using finite element method for spatial discretization. The combined two stages lead to a Lagrange–Galerkin method which is robust, second order accurate, and simple to implement for problems on complex geometry. Numerical results are shown for several test problems with different ranges of difficulty.