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The purpose of this paper is to introduce efficient methods for solving the 2D biharmonic equation with Dirichlet boundary conditions of second kind. This equation describes the deflection of loaded plate with boundary conditions of simply supported plate kind. Also it can be derived from the calculus of variations combined with the variational principle of minimum potential energy. Because of existing fourth derivatives in this equation, introducing high‐order accurate methods need to use artificial points. Also solving the resulted linear system of equations suffers from slow convergence when iterative methods are used. This paper aims to introduce efficient methods to overcome these problems.

The paper considers several compact finite difference approximations that are derived on a nine‐point compact stencil using the values of the solution and its second derivatives as the unknowns. In these approximations there is no need to define special formulas near the boundaries and boundary conditions can be incorporated with these techniques. Several iterative linear systems solvers such as Krylov subspace and multigrid methods and their combination (with suitable preconditioner) have been developed to compare the efficiency of each method and to design powerful solvers.

The paper finds that the combination of compact finite difference schemes with multigrid method and Krylov iteration methods preconditioned by multigrid have excellent results for the second biharmonic equation, and that Krylov iteration methods preconditioned by multigrid are the most efficient methods.

The paper is of value in presenting, via some tables and figures, some numerical experiments which resulted from applying new methods on several test problems, and making comparison with conventional methods.

The centrifugal instability mechanism of boundary layers over concave surfaces is responsible for the development of quasi-periodic, counter-rotating vortices aligned in a streamwise direction known as Görtler vortices. By distorting the boundary layer structure in both the spanwise and the wall-normal directions, Görtler vortices may modify heat transfer rates. The purpose of this study is to conduct spatial numerical simulation experiments based on a vorticity–velocity formulation of the incompressible Navier–Stokes system of equations to quantify the role of the transition in the heat transfer process.

Experiments are conducted using an in-house, parallel, message-passing code. Compact finite difference approximations and a spectral method are used to approximate spatial derivatives. A fourth-order Runge–Kutta method is adopted for time integration. The Poisson equation is solved using a geometric multigrid method.

Results show that the numerical method can capture the physics of transitional flows over concave geometries. They also show that the heat transfer rates in the late stages of the transition may be greater than those for either laminar or turbulent ones.

The numerical method can be considered as a robust alternative to investigate heat transfer properties in transitional boundary layer flows over concave surfaces.

The purpose of this paper is to obtain analytic solutions of telegraph equation by the homotopy Padé method.

The authors used Maple Package to calculate the solutions obtained from the homotopy Padé method.

The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m, m] homotopy Padé technique are often independent of auxiliary parameter h and this technique accelerates the convergence of the related series. Finally, numerical results for some test problems with known solutions are presented and the numerical results are given to show the efficiency of the proposed techniques.

The paper is shown that homotopy Padé technique is a promising tool with accelerated convergence for complicated nonlinear differential equations.

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

From a theoretical point of view the question of spurious modes has been regarded as a closed problem. However, in this paper we show that even a precise definition of spurious‐free approximation was lacking. Hence, a sound definition of spurious‐free finite element method is given and a set of necessary and sufficient conditions ensuring that a finite element method is spurious‐free in the defined sense is stated. A critical comparison between the proposed theory and the currently accepted one is then carried out and existing counterexamples to the latter are pointed out. Comparison with an older theory leads to another set of necessary and sufficient conditions providing a better grasp of the key feature a finite element space must have to rule out spurious modes. The impact of the proposed theory is stressed, showing that Nedelec's tetrahedral edge elements of all orders provide spurious‐free approximations in all conditions of practical interest. Finally, it is shown, for the first time to the best of authors’ knowledge, that also many high‐order edge elements, recently proposed in the engineering literature for the analysis of electromagnetic problems, provide the same kind of reliable approximation.

Variable density flows play an important role in many technological devices and natural phenomena. The purpose of this paper is to develop a robust and accurate method for low Mach number flows with large density and temperature variations.

Low Mach number approximation approach is used in the paper combined with a predictor-corrector method and accurate compact scheme of fourth and sixth order. A novel algorithm is formulated for the projection method in which the boundary conditions for the pressure are implemented in such a way that the continuity equation is fulfilled everywhere in the computational domain, including the boundary nodes.

It is shown that proposed implementation of the boundary conditions considerably improves a solution accuracy. Assessment of the accuracy was performed based on the constant density Burggraf flow and for two benchmark cases for the natural convection problems: steady flow in a square cavity and unsteady flow in a tall cavity. In all the cases the results agree very well with exemplary solutions.

A staggered or half-staggered grid arrangement is usually used for the projection method for both constant and low Mach number flows. The staggered approach ensures stability and strong pressure-velocity coupling. In the paper a high-order compact method has been implemented in the framework of low Mach number approximation on collocated meshes. The resulting algorithm is accurate, robust for large density variations and is almost free from the pressure oscillations.

The purpose of this paper is to present a new discretisation scheme, based on equation-coupled approach and high-order five-point integrated radial basis function (IRBF) approximations, for solving the first biharmonic equation, and its applications in fluid dynamics.

The first biharmonic equation, which can be defined in a rectangular or non-rectangular domain, is replaced by two Poisson equations. The field variables are approximated on overlapping local regions of only five grid points, where the IRBF approximations are constructed to include nodal values of not only the field variables but also their second-order derivatives and higher-order ones along the grid lines. In computing the Dirichlet boundary condition for an intermediate variable, the integration constants are used to incorporate the boundary values of the first-order derivative into the boundary IRBF approximation.

These proposed IRBF approximations on the stencil and on the boundary enable the boundary values of the derivative to be exactly imposed, and the IRBF solution to be much more accurate and not influenced much by the RBF width. The error is reduced at a rate that is much greater than four. In fluid dynamics applications, the method is able to capture well the structure of steady highly non-linear fluid flows using relatively coarse grids.

The main contribution of this study lies in the development of an effective high-order five-point stencil based on IRBFs for solving the first biharmonic equation in a coupled set of two Poisson equations. A fast rate of convergence (up to 11) is achieved.

The purpose of this paper is to both determine the effects of the nonlinearity on the wave dynamics and assess the temporal and spatial accuracy of five finite difference methods for the solution of the inviscid generalized regularized long-wave (GRLW) equation subject to initial Gaussian conditions.

Two implicit second- and fourth-order accurate finite difference methods and three Runge-Kutta procedures are introduced. The methods employ a new dependent variable which contains the wave amplitude and its second-order spatial derivative. Numerical experiments are reported for several temporal and spatial step sizes in order to assess their accuracy and the preservation of the first two invariants of the inviscid GRLW equation as functions of the spatial and temporal orders of accuracy, and thus determine the conditions under which grid-independent results are obtained.

It has been found that the steepening of the wave increase as the nonlinearity exponent is increased and that the accuracy of the fourth-order Runge-Kutta method is comparable to that of a second-order implicit procedure for time steps smaller than 100th, and that only the fourth-order compact method is almost grid-independent if the time step is on the order of 1,000th and more than 5,000 grid points are used, because of the initial steepening of the initial profile, wave breakup and solitary wave propagation.

This is the first study where an accuracy assessment of wave breakup of the inviscid GRLW equation subject to initial Gaussian conditions is reported.

The diffusion‐advection phenomena occur in many physical situations such as, the transport of heat in fluids, flow through porous media, the spread of contaminants in fluids and as well as in many other branches of science and engineering. So it is essential to approximate the solution of these kinds of partial differential equations numerically in order to investigate the prediction of the mathematical models, as the exact solutions are usually unavailable.

The difficulties arising in numerical solutions of the transport equation are well known. Hence, the study of transport equation continues to be an active field of research. A number of mathematicians have developed the method of time‐splitting to divide complicated time‐dependent partial differential equations into sets of simpler equations which could then be solved separately by numerical means over fractions of a time‐step. For example, they split large multi‐dimensional equations into a number of simpler one‐dimensional equations each solved separately over a fraction of the time‐step in the so‐called locally one‐dimensional (LOD) method. In the same way, the time‐splitting process can be used to subdivide an equation incorporating several physical processes into a number of simpler equations involving individual physical processes. Thus, instead of applying the one‐dimensional advection‐diffusion equation over one time‐step, it may be split into the pure advection equation and the pure diffusion equation each to be applied over half a time‐step. Known accurate computational procedures of solving the simpler diffusion and advection equations may then be used to solve the advection‐diffusion problem.

In this paper, several different computational LOD procedures were developed and discussed for solving the two‐dimensional transport equation. These schemes are based on the time‐splitting finite difference approximations.

The new approach is simple and effective. The results of a numerical experiment are given, and the accuracy are discussed and compared.

A comparison of calculations with the results of the conventional finite difference techniques demonstrates the good accuracy of the proposed approach.

The purpose of this paper is to present a new hierarchical finite element formulation for approximation in time.

The present approach using wavelets as basis functions provides a global control over the solution error as the equation of motion is satisfied for the entire duration in the weighted integral sense. This approach reduces the semi‐discrete system of equations in time to be solved to a single algebraic problem, in contrast to step‐by‐step time integration methods, where a sequence of algebraic problems are to be solved to compute the solution.

The proposed formulation has been validated for both inertial and wave propagation types of problems. The stability and accuracy characteristics of the proposed formulation has been examined and is found to be energy conserving.

The paper presents a new hierarchical finite element formulation for the solution of structural dynamics problems. This formulation uses wavelets as the analyzing basis for the desired transient solution. It is found to be very well behaved in solution of wave‐propagation problems.