Box Scheme Research Articles

Some implicit methods for the numerical solution of Burgers’ equation

In this article we develop multisymplectic box methods on Burgers’ equation. Burgers’ equation has some discontinuous solutions because of effects of viscosity term. These discontinuities raise phenomenon of shock waves. Some explicit and implicit numerical methods have been experimented on Burgers’ equation but these schemes have not been seen proper in this case because of their conditional stability and existence of viscosity term. To avoid these problems, multisymplectic box scheme which has been constructed for KdV equation is purposed for Burgers’ equation. Multisymplectic box scheme is a very effective box scheme in diminishing artificial wiggles which appear in approximation solutions. We consider two types of box schemes and implement on Burgers’ equation to get better results with no artificial wiggles.

Multisymplectic Euler Box Scheme for the KdV Equation

We investigate the multisymplectic Euler box scheme for the Korteweg–deVries (KdV) equation. A new completely explicit six-point scheme isderived. Numerical experiments of the new scheme with comparisons tothe Zabusky–Kruskal scheme, the multisymplectic 12-point scheme, thenarrow box scheme and the spectral method are made to show nicenumerical stability and ability to preserve the integral invariant forlong-time integration.

A trainable feature extractor for handwritten digit recognition

This article focuses on the problems of feature extraction and the recognition of handwritten digits. A trainable feature extractor based on the LeNet5 convolutional neural network architecture is introduced to solve the first problem in a black box scheme without prior knowledge on the data. The classification task is performed by support vector machines to enhance the generalization ability of LeNet5. In order to increase the recognition rate, new training samples are generated by affine transformations and elastic distortions. Experiments are performed on the well-known MNIST database to validate the method and the results show that the system can outperform both SVMs and LeNet5 while providing performances comparable to the best performance on this database. Moreover, an analysis of the errors is conducted to discuss possible means of enhancement and their limitations.

The Preissmann box scheme and its modification for transcritical flows

AbstractThe Preissmann box scheme is the standard numerical method used by hydraulic engineers to model open channel flows or surcharged flows in pipes; but it breaks down for transcritical flows. Various alternatives have been suggested for such cases, but we show how simple modifications of the difference scheme and its method of solution enable both to be used for such flows. Analysis of the double‐sweeping Thomas algorithm shows both how it works in the subcritical case and the effectiveness of its modification for transcritical flows. A well‐known analytical transcritical flow is used to demonstrate the accuracy of the new numerical scheme. Copyright © 2006 John Wiley & Sons, Ltd.

Researching European ‘alternative’ food networks: some methodological considerations

Recent European literature on ‘alternative’ food networks (AFNs) draws heavily upon an apparently accessible and diverse body of non‐conventional food networks in the agro‐food sector and whilst researchers frequently refer to individual examples of farmers markets, box schemes, producer cooperatives and community‐supported agriculture projects, less attention is given to the methodological processes that facilitate the identification and examination of these networks. From the preliminary stages of a research project focusing on examples of AFNs, 2 this paper examines the process of operationalizing AFNs research and reviews the difficulties associated with identifying, comparing and characterizing AFNs.

Magnetohydrodynamic natural convection flow on a sphere with uniform heat flux in presence of heat generation

Magnetohydrodynamic natural convection boundary layer flow on a sphere with uniform heat flux in presence of heat generation has been investigated in this paper. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations is then solved numerically by two distinct efficient methods, namely (i) implicit finite difference method together with the Keller box scheme and (ii) perturbation or series solution technique. The results of the surface shear stress in terms of local skin friction and the rate of heat transfer in terms of local Nusselt number, velocity distribution as well as temperature distribution are shown graphically for a selection of parameter sets consisting of the heat generation parameter and the magnetic parameter.

A new multisymplectic scheme for generalized Kadomtsev-Petviashvili equation

From the multisymplectic Euler box scheme, we derive a new multisymplectic 16-point scheme for the generalized Kadomtsev-Petviashvili equation by eliminating the auxiliary variables. The new scheme is an explicit scheme in sense that it does not need iteration. The results of backward error analysis for the Euler box scheme show that the modified equation associated with the new scheme is not multisymplectic but of a high order approximation to a multisymplectic partial differential equations with a modified structure. Numerical results on one soliton and lump-type solitary waves are reported to illustrate the efficiency of the multisymplectic scheme.

Natural convection flow from an isothermal horizontal circular cylinder in presence of heat generation

Natural convection laminar boundary layer flow from a horizontal circular cylinder with a uniform surface temperature at presence of heat generation has been investigated. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear systems of partial differential equations are solved numerically applying two distinct methods namely (i) implicit finite difference method together with the Keller box scheme and (ii) series solution technique. The results of the surface shear stress in terms of the local skin friction and the surface rate of heat transfer in terms of the local Nusselt number for a selection of the heat generation parameter γ (= 0.0, 0.2, 0.5, 0.8, 1.0) are obtained and presented in both tabular and graphical formats. Without effect of the internal heat generation inside the fluid domain for which we take γ = 0.0, the present numerical results show an excellent agreement with those of Merkin [J.H. Merkin, Free convection boundary layer on an isothermal horizontal circular cylinders, in: ASME/AIChE, Heat Transfer Conference, St. Louis, MO, August 9–11, 1976]. The effects of γ on the fluid velocity, temperature distribution, streamlines and isotherms are examined.

The Shifted Box Scheme for Scalar Transport Problems

When solving certain scalar transport problems, it is important to use fast methods that do not produce any parasitic oscillating solutions, but still have good energy conservation properties. The well known box scheme has these properties, but the use is restricted by severe conditions on the sign of the coefficients. In order to avoid this restriction, we introduce a modified method, that we call the shifted box scheme. It is very efficient for initial-boundary value problems, since it does not require more work per time step than an explicit scheme, while still being unconditionally stable.

Recent advances in finite element methods for multi-phase flow processes in porous media

In this paper, we will focus on three essentially important issues considering the modeling of multi-phase flow in heterogeneous porous media. Depending on the governing processes, the multi-phase flow equation system can show either predominant advection or diffusion/dispersion. Numerical methods have to be able to capture both cases. Furthermore, realistic porous media are always heterogeneous. The physically correct description of the interfacial behaviour is numerically very challenging, as discontinuities in primary variables may occur. In this paper, we review these issues and present the box scheme, a control volume finite element method, including an interface condition as one numerical method capable of tackling the three problems. In the end, we will give the results of a 3D simulation.

Nonconforming finite element methods with subgrid viscosity applied to advection‐diffusion‐reaction equations

AbstractA nonconforming (Crouzeix–Raviart) finite element method with subgrid viscosity is analyzed to approximate advection‐diffusion‐reaction equations. The error estimates are quasi‐optimal in the sense that keeping the Péclet number fixed, the estimates are suboptimal of order ${1\over 2}$ in the mesh size for the L2‐norm and optimal for the advective derivative on quasi‐uniform meshes. The method is also reformulated as a finite volume box scheme providing a reconstruction formula for the diffusive flux with local conservation properties. Numerical results are presented to illustrate the error analysis. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006

The exponential method of angled derivatives

A hybrid numerical method is presented for a linear, first order, hyperbolic partial differential equation (PDE): the nonhomogeneous one-way advection-reaction equation. This PDE is first reduced to an ordinary differential equation (ODE) along a characteristic emanating backward in time from each mesh point. These ODEs are then integrated numerically using exponential fitting to advance in time thus permitting wide disparities in size among the coefficients. Such temporal integration requires a spatial interpolation procedure dependent upon the slope of the local characteristic. If the local Courant number is less than one then the interpolation is afforded by the exponential angled derivative method while if greater than one then the exponential reflected angled derivative approximation is employed. Otherwise, if the Courant number is transitioning through one then the exponential box scheme is employed. At the boundary, parabolic tracing of the characteristics provides a comparable level of accuracy. The net result is a three-level explicit method which is second-order accurate and unconditionally stable. The efficacy of this exponential method of angled derivatives (EMAD) scheme for singular perturbation problems is demonstrated numerically.

Analysis of Box Schemes for Reactive Flow Problems

Key properties of the box scheme are shown to be advantageous for reactive flow problems. Unconditional stability and compact conservation are shown by a detailed modified equation analysis to enable the scheme to reflect exactly the reduced speed, enhanced diffusion, and dispersion which are typical of such hyperbolic conservation laws with relaxation. A novel modified equation analysis is also used to show how the spurious checkerboard mode behaves and can be controlled. Numerical experiments for some nonlinear one-dimensional problems and a two-dimensional problem demonstrate that the behavior of the scheme deduced from a simple model problem has general validity.

Mixed Convection Boundary Layer Flow of a Micropolar Fluid Along a Vertical Cylinder

The steady mixed convection boundary layer flow of a micropolar fluid along a vertical cylinder is studied for isothermal as well as uniform surface heat flux boundary conditions. The transformed conservation equations of the nonsimilar boundary layers are solved numerically using the Keller box scheme. Numerical solutions for the velocity, angular velocity, and temperature fields are presented for different values of the material and mixed convection parameters. The wall shear stress and the surface heat transfer rate increase with buoyancy and cylindrical curvature.

On Symplectic and Multisymplectic Schemes for the KdV Equation

We examine some symplectic and multisymplectic methods for the notorious Korteweg–de Vries equation, with the question whether the added structure preservation that these methods offer is key in providing high quality schemes for the long time integration of nonlinear, conservative partial differential equations. Concentrating on second order discretizations, several interesting schemes are constructed and studied. Our essential conclusions are that it is possible to design very stable, conservative difference schemes for the nonlinear, conservative KdV equation. Among the best of such schemes are methods which are symplectic or multisymplectic. Semi-explicit, symplectic schemes can be very effective in many situations. Compact box schemes are effective in ensuring that no artificial wiggles appear in the approximate solution. A family of box schemes is constructed, of which the multisymplectic box scheme is a prominent member, which are particularly stable on coarse space–time grids

Natural convection flow from an isothermal horizontal circular cylinder with temperature dependent viscosity

Free convection flow from an isothermal horizontal circular cylinder immersed in a fluid with viscosity proportional to an inverse linear function of temperature is studied. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations is reduced to local non-similarity equations which are solved numerically by a very efficient implicit finite difference method together with Keller box scheme. Numerical results are presented by velocity and viscosity profiles of the fluid as well as heat transfer characteristics, namely the local heat transfer rate and the local skin-friction coefficients for a wide range of viscosity parameter e (= 0.0, 0.5, 1.0, 2.0, 3.0,4.0) and the Prandtl number Pr (= 1.0, 7.0, 10.0, 15.0, 20.0, 30.0).

Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs

Several recently developed multisymplectic schemes for Hamiltonian PDEs have been shown to preserve associated local conservation laws and constraints very well in long time numerical simulations. Backward error analysis for PDEs, or the method of modified equations, is a useful technique for studying the qualitative behavior of a discretization and provides insight into the preservation properties of the scheme. In this paper we initiate a backward error analysis for PDE discretizations, in particular of multisymplectic box schemes for the nonlinear Schrödinger equation. We show that the associated modified differential equations are also multisymplectic and derive the modified conservation laws which are satisfied to higher order by the numerical solution. Higher order preservation of the modified local conservation laws is verified numerically.

The method of angled derivatives

A hybrid numerical method is presented for a linear, first order, hyperbolic partial differential equation (PDE): the nonhomogeneous one-way advection equation. The unifying thread is the reduction of the PDE to an ordinary differential equation (ODE) along a characteristic emanating backward in time from each mesh point. These ODEs are then integrated numerically to advance in time. Such integration requires an interpolation procedure dependent upon the slope of the local characteristic. If the local Courant number is less than one then the interpolation is afforded by the angled derivative method while if greater than one then a reflected angled derivative approximation is introduced. Otherwise, if the Courant number is transitioning through one then the box scheme is employed. At the boundary, parabolic tracing of the characteristics provides a comparable level of accuracy. The net result is a compact, nondissipative, three-level explicit method which is second-order accurate and unconditionally stable. The efficacy of this method of angled derivatives (MAD) scheme for both constant and variable coefficient problems, with or without inhomogeneities, is demonstrated numerically.

Analysis of transient flow in pipelines with fluid-structure interaction using method of lines

A mathematical model is presented for transient flow in a pipeline with fluid–structure interaction. Water hammer theory and equations of axial motion for the pipeline are employed and the Poisson, junction and transient shear stress couplings are taken into account, which give rise to four coupled non-linear, first-order hyperbolic partial differential equations governing the fluid flow and pipe motion. These equations are discretized in space using the Keller box scheme and the method of lines is employed to reduce the partial differential equations to a system of ordinary differential equations. The resulting system is solved using a backward differentiation formulation method. The effect of transient shear stress on transient flow is investigated and the mechanisms underlying this effect are explored. The results revealed that the influence of transient shear stress can be significant and varies considerably, depending on the boundary conditions, viz, valve closure time. Copyright © 2005 John Wiley & Sons, Ltd.

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