A study of a numerical solution of the steady two dimensions Navier-Stokes equations in a constricted channel problem by a compact fourth order method

We present a numerical solution for the steady 2D Navier-Stokes equations using a fourth order compact-type method. The geometry of the problem is a constricted symmetric channel, where the boundary can be varied, via a parameter, from a smooth constriction to one possessing a very sharp but smooth corner allowing us to analyse the behaviour of the errors when the solution is smooth or near singular. The set of non-linear equations is solved by the Newton method. Results have been obtained for Reynolds number up to 500. Estimates of the errors incurred have shown that the results are accurate and better than those of the corresponding second order method. (C) 2002 Elsevier B.V. All rights reserved.

Fourth-order method for solving the Navier-Stokes equations in a constricting channel

A fourth-order numerical method for solving the Navier-Stokes equations in streamfunction/vorticity formulation on a two-dimensional non-uniform orthogonal grid has been tested on the fluid flow in a constricted symmetric channel. The family of grids is generated algebraically using a conformal transformation followed by a non-uniform stretching of the mesh cells in which the shape of the channel boundary can vary from a smooth constriction to one which one possesses a very sharp but smooth corner. The generality of the grids allows the use of long channels upstream and downstream as well as having a refined grid near the sharp corner. Derivatives in the governing equations are replaced by fourth-order central differences and the vorticity is eliminated, either before or after the discretization, to form a wide difference molecule for the streamfunction. Extra boundary conditions, necessary for wide-molecule methods, are supplied by a procedure proposed by Henshaw et al. The ensuing set of non-linear equations is solved using Newton iteration. Results have been obtained for Reynolds numbers up to 250 for three constrictions, the first being smooth, the second having a moderately sharp corner and the third with a very sharp corner. Estimates of the error incurred show that the results are very accurate and substantially better than those of the corresponding second-order method. The observed order of the method has been shown to be close to four, demonstrating that the method is genuinely fourth-order. © 1977 John Wiley & Sons, Ltd.

A QUADRATIC C degrees INTERIOR PENALTY METHOD FOR LINEAR FOURTH ORDER BOUNDARY VALUE PROBLEMS WITH BOUNDARY CONDITIONS OF THE CAHN-HILLIARD TYPE

We develop a quadratic C degrees interior penalty method for linear fourth order boundary value problems with essential and natural boundary conditions of the Cahn-Hilliard type. Both a priori and a posteriori error estimates are derived. The performance of the method is illustrated by numerical experiments.

Analysis of an Interior Penalty Method for Fourth Order Problems on Polygonal Domains

Error analysis for a stable C (0) interior penalty method is derived for general fourth order problems on polygonal domains under minimal regularity assumptions on the exact solution. We prove that this method exhibits quasi-optimal order of convergence in the discrete H (2), H (1) and L (2) norms. L (a) norm error estimates are also discussed. Theoretical results are demonstrated by numerical experiments.

A C-0 Interior Penalty Method for a Fourth-order Variational Inequality of the Second Kind

In this article, we propose a C-0 interior penalty ((CIP)-I-0) method for the frictional plate contact problem and derive both a priori and a posteriori error estimates. We derive an abstract error estimate in the energy norm without additional regularity assumption on the exact solution. The a priori error estimate is of optimal order whenever the solution is regular. Further, we derive a reliable and efficient a posteriori error estimator. Numerical experiments are presented to illustrate the theoretical results. (c) 2015Wiley Periodicals, Inc.

Fourth order accurate compact scheme with group velocity control (GVC)

For solving complex flow field with multi-scale structure higher order accurate schemes are preferred. Among high order schemes the compact schemes have higher resolving efficiency. When the compact and upwind compact schemes are used to solve aerodynamic problems there are numerical oscillations near the shocks. The reason of oscillation production is because of non-uniform group velocity of wave packets in numerical solutions. For improvement of resolution of the shock a parameter function is introduced in compact scheme to control the group velocity. The newly developed method is simple. It has higher accuracy and less stencil of grid points.

Solutions of fourth-order parabolic equation modeling thin film growth

In this paper we study the well-posedness for a fourth-order parabolic equation modeling epitaxial thin film growth. Using Kato's Method [1], [2] and [3] we establish existence, uniqueness and regularity of the solution to the model, in suitable spaces, namelyC⁰([0,T];L^p(Ω)) where  with 1<α<2, n∈N and n≥2. We also show the global existence solution to the nonlinear parabolic equations for small initial data. Our main tools are L_p–L_q-estimates, regularization property of the linear part of e^−tΔ2 and successive approximations. Furthermore, we illustrate the qualitative behavior of the approximate solution through some numerical simulations. The approximate solutions exhibit some favorable absorption properties of the model, which highlight the stabilizing effect of our specific formulation of the source term associated with the upward hopping of atoms. Consequently, the solutions describe well some experimentally observed phenomena, which characterize the growth of thin film such as grain coarsening, island formation and thickness growth.

GA-based Optimization of a Fourth-order Sigma-delta Modulator for WLAN

Over-sampling sigma-delta analogue-to-digital converters (ADCs) are one of the key building blocks of state of the art wireless transceivers. In the sigma-delta modulator design the scaling coefficients determine the overall signal-to-noise ratio. Therefore, selecting the optimum value of the coefficient is very important. To this end, this paper addresses the design of a fourthorder multi-bit sigma-delta modulator for Wireless Local Area Networks (WLAN) receiver with feed-forward path and the optimum coefficients are selected using genetic algorithm (GA)- based search method. In particular, the proposed converter makes use of low-distortion swing suppression SDM architecture which is highly suitable for low oversampling ratios to attain high linearity over a wide bandwidth. The focus of this paper is the identification of the best coefficients suitable for the proposed topology as well as the optimization of a set of system parameters in order to achieve the desired signal-to-noise ratio. GA-based search engine is a stochastic search method which can find the optimum solution within the given constraints.

Monotone positive solutions for a fourth order equation with nonlinear boundary conditions

This work is concerned with the existence of monotone positive solutions for a class of beam equations with nonlinear boundary conditions. The results are obtained by using the monotone iteration method and they extend early works on beams with null boundary conditions. Numerical simulations are also presented. (C) 2009 Elsevier Ltd. All rights reserved.

Effects of fourth-order fiber dispersion on ultrashort parabolic optical pulses in the normal dispersion regime

We propose a new method for the generation of both triangular-shaped optical pulses and flat-top, coherent supercontinuum spectra using the effect of fourth-order dispersion on parabolic pulses in a passive, normally dispersive highly nonlinear fiber. The pulse reshaping process is described qualitatively and is compared to numerical simulations.

Effects of fourth-order fiber dispersion on ultrashort parabolic optical pulses in the normal dispersion regime

We propose a new method for the generation of both triangular-shaped optical pulses and flat-top, coherent supercontinuum spectra using the effect of fourth-order dispersion on parabolic pulses in a passive, normally dispersive highly nonlinear fiber. The pulse reshaping process is described qualitatively and is compared to numerical simulations.

Adaptive mesh generation for singularly perturbed fourth-order ordinary differential equations

In this paper, we consider a singularly perturbed boundary-value problem for fourth-order ordinary differential equation (ODE) whose highest-order derivative is multiplied by a small perturbation parameter. To solve this ODE, we transform the differential equation into a coupled system of two singularly perturbed ODEs. The classical central difference scheme is used to discretize the system of ODEs on a nonuniform mesh which is generated by equidistribution of a positive monitor function. We have shown that the proposed technique provides first-order accuracy independent of the perturbation parameter. Numerical experiments are provided to validate the theoretical results.