A Modified Trefftz Method for Two-Dimensional Laplace Equation Considering the Domain's Characteristic Length

Exact boundary condition for time-dependent wave equation based on boundary integral

This paper briefly describes the development of generalized collocation methods for solving coupled systems of nonlinear, one-dimensional, parabolic, partial differential equations. parabolic, partial differential equations. The methods are applied to several problems that are representative of fluid flow in a porous medium. The numerical results for the problems solved indicate that collocation methods using C piecewise polynomials with Gauss-Legendre collocation points polynomials with Gauss-Legendre collocation points are more efficient than conventional, second-order, finite-difference methods. In particular, for problems whose solutions are smooth, one obtains more accuracy per unit of computer time with increasingly higher-order collocation methods. For problems whose solutions are characterized by a steep, shock-like, transient wave front, one can conclude that increasing the order of the collocation method does not necessarily result in a more efficient method, but high-accuracy collocation methods still appear to be more efficient than conventional, second-order, finite-difference methods. Also, for a given accuracy, collocation methods generally require less computer storage than conventional, second-order, finite-difference methods. Finally, the collocation technique presented here is shown to be not directly applicable to the saturation equation for immiscible flow. Introduction The purpose of this paper is to investigate collocation methods for the numerical solution of nonlinear equations similar to those that describe the flow of a fully compressible fluid in a porous medium and those that describe the process by which one miscible liquid displaces another liquid in a porous medium. The objective is to determine if collocation methods are applicable and economically justified for solving these types of petroleum engineering problems. The conclusions petroleum engineering problems. The conclusions reached are based on the amount of computer time and computer storage that is required to obtain a specified accuracy. Since only two problems are solved, the results may not be extendable to more general problems; however, experience in solving a wide variety of problems indicates that this is the general behavior that might be expected when using collocation methods. Previously published results indicate how Galerkin, collocation, and finite-element methods can be used to solve various problems that arise in petroleum engineering. Most of these papers present a variety of arguments in favor of these present a variety of arguments in favor of these methods. The principal argument is that better answers can be obtained for the same computational effort than by finite-difference methods. Culham and Varga have presented the most convincing evidence in favor of Galerkin methods. However, they only considered piecewise linear and piecewise cubic-basis functions and they only considered one problem. Because of these limitations, it is not problem. Because of these limitations, it is not clear whether their conclusions can be extended to higher-order Galerkin methods or to other types of problems. Also, they did not consider collocation problems. Also, they did not consider collocation methods in any detail. This paper considers the use of increasingly higher-order collocation methods and compares the results with those obtained with the usual second-order, finite-difference method. The Galerkin method is not considered, even though that is the method with which collocation must ultimately compete. This was done for several reasons. A preliminary operation count indicates that Galerkin methods will not be competitive with collocation methods for most problems. Also, it is not clear how to implement the higher-order Galerkin methods to achieve maximum efficiency. These remarks indicate that the relative merits of collocation and Galerkin methods are still uncertain. PROBLEMS SOLVED PROBLEMS SOLVED Two problems are solved to serve as a basis for the comparisons. The first problem, called the "gas flow problem," is the same problem considered by Culham and Varga. This problem is nonlinear and includes a volumetric source term. SPEJ P. 345

For three-dimensional simulations of the vortex phenomena in superconductors based on the Ginzburg--Landau (GL) theory, the physical variables must be solved in the whole space in general. Exact boundary conditions on an artificial boundary are discussed to reformulate the problem in a finite domain. Approximations of these boundary conditions with high accuracy are given, and their convergence properties are examined. Error estimates are derived for the finite element approximations with the approximate boundary conditions.

The problem of imposing accurate radiation boundary conditions at the artificial boundary is generally the primary accuracy bottleneck in the numerical solution of scattering problems in the time domain. An impressive number of schemes have been proposed to overcome this difficulty. Nonetheless, the ideal of solving the problem on a fixed domain and achieving convergence by some systematic improvement of the boundary treatment and/or simple mesh refinement seems rarely to have been used. In this talk we advocate such an approach for the basic constant coefficient equations of wave propagation — the wave equation, Maxwell’s equations, the convective wave equation and the compressible Euler equations linearized about a uniform flow. We first develop representations of the exact boundary conditions for special boundaries using separation of variables. The difficult, that is temporally nonlocal, part of the exact condition is expressed as a temporal convolution operator. Approximate boundary treatments, both new and old, are viewed as approximations to the exact convolution kernel, allowing a fairly easy identification of convergence rates and the development of complexity estimates.

A novel Trefftz method for solving the multi-dimensional direct and Cauchy problems of Laplace equation in an arbitrary domain

A variant of Trefftz's method by boundary Fourier expansion for solving regular and singular plane boundary-value problems

The flow of a liquid film over an inclined plane is frequently found in nature and industry, and, under some conditions, instabilities in the free surface may appear. These instabilities are initially two-dimensional surface waves, known as Kapitza waves. Surface waves are important to many industrial applications. For example, liquid films with surface waves are employed to remove heat from solid surfaces. The initial phase of the instability is governed by the Orr-Sommerfeld equation and the appropriate boundary conditions; therefore, the fast and accurate solution of this equation is useful for industry. This paper presents a spectral method to solve the Orr-Sommerfeld equation with free surface boundary conditions. Our numerical approach is based on a Galerkin method with Chebyshev polynomials of the first kind, making it possible to express the Orr-Sommerfeld equation and their boundary conditions as a generalized eigenvalue problem. The main advantages of the present spectral method when compared to others such as, for instance, spectral collocation, are its stability and its readiness in including the boundary conditions in the discretized equations. We compare our numerical results with analytical solutions based on Perturbation Methods, which are valid only for long wave instabilities, and show that the results agree in the region of validity of the long-wave hypothesis. Far from this region, our results are still valid. In addition, we compare our results with published experimental results and the agreement is very good. The method is stable, fast, and capable to solve initial instabilities in free surface flows.

The numerical solution of the time dependent wave equation in an unbounded domain generally leads to a truncation of this domain, which requires the introduction of an artificial boundary with associated boundary conditions. Such nonreflecting conditions ensure the equivalence between the solution of the original problem in the unbounded region and the solution inside the artificial boundary. We consider the acoustic wave equation and derive exact transparent boundary conditions that are local in time and can be directly used in explicit methods. These conditions annihilate wave harmonics up to a given order on a spherical artificial boundary, and we show how to combine the derived boundary condition with a finite difference method. The analysis is complemented by a numerical example in two spatial dimensions that illustrates the usefulness and accuracy of transparent boundary conditions.

\nThe paper presents mathematical modelling of flow boiling heat transfer in a rectangular minichannel asymmetrically heated by a thin and one-sided enhanced foil. Both surfaces are available for observations due to the openings covered with glass sheets. Thus, changes in the colour of the plain foil surface can be registered and then processed. Plain side of the heating foil is covered with a base coat and liquid crystal paint. Observation of the opposite, enhanced surface of the minichannel allows for identification of the gas-liquid two-phase flow patterns and vapour quality. A two-dimensional mathematical model of heat transfer in three subsequent layers (sheet glass, heating foil, liquid) was proposed. Heat transfer in all these layers was described with the respective equations: Laplace equation, Poisson equation and energy equation, subject to boundary conditions corresponding to the observed physical process. The solutions (temperature distributions) in all three layers were obtained by Trefftz method. Additionally, the temperature of the boiling liquid was obtained by homotopy perturbation method (HPM) combined with Trefftz method. The heat transfer coefficient, derived from Robin boundary condition, was estimated in both approaches. In comparison, the results by both methods show very good agreement especially when restricted to the thermal sublayer.\n

A comparison of techniques for local stability analysis of tubular reactor systems

The Laplace equation was solved in the interelectrode space for shaped electrodes (two-dimensional case) by the method of finite differences (FDM), Galerkin's method (GM), and collocation method (CM). A comparison shows that for electrode shapes with a continuously changing surface (continuous first and second derivatives), the solutions by all three methods are equivalent, giving identical distribution of local current densities on the electrode surface. The use of GM and CM is, however, not practical because of high requirements on the computer time and memory as compared with FDM. Moreover, the GM and CM fail in the case of discontinuities in the electrode shape.

We present a new and very powerful theoretical method-a distorted-wave Born approximation (DWBA) with an arbitrary final-state electron-electron correlation function. This method combines the flexibility of including any theoretically desired electron-electron correlation and the physical clarity of the DWBA method. Calculations explore the use of an auxiliary final-state wave function with the exact boundary condition, in attempting to describe the absolute cross sections for ionisation of helium in coplanar symmetric kinematics and to resolve discrepancies between theory and experiment.

Artificial boundary conditions and finite difference approximations for a time-fractional diffusion-wave equation on a two-dimensional unbounded spatial domain

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

An asymptotic local solution for axi-symmetric heat conduction problems with temperature discontinuity on the boundary