An implicit,compact, finite difference method to solve hyperbolic equations

At present most approximate (discrete) solutions of time dependent hyperbolic equations are obtained by explicit finite difference methods, where the maximal allowable time step is given by a condition of numerical stability (i.e., the CFL condition). This report contains the development and the analysis of an implicit method of high order accuracy which is unconditionally stable, thus allowing to progress much faster in time. Furthermore, the presence of additional artificial boundary conditions does have an influence on the accuracy of the solution but not on the stability of the method. The numerical scheme has been checked considering two examples: the solution of the (linear) wave equation and the non linear Euler equations of fluid dynamics.     

Convergence of finite difference transient response computations for thin shells

Numerical studies pertaining to the limits of applicability of the finite difference method in the solution of linear transient shell response problems are performed, and a computational procedure for the use of the method is recommended. It is found that the only inherent limitation of the finite difference method is its inability to reproduce accurately response discontinuities. This is not a serious limitation in view of natural constraints imposed by the extension of Saint Venant's principle to transient response problems. It is also found that the short wavelength limitations of thin shell (Bernoulli-Euler) theory create significant convergence difficulties in computed response to certain types of transverse excitations. These difficulties may be overcome, however, through proper selection of finite difference mesh dimensions and temporal smoothing of the excitation.     

A discontinuous finite difference streamline diffusion method for time-dependent hyperbolic problems

In this article, a new finite element method, discontinuous finite difference streamline diffusion method (DFDSD), is constructed and studied for first-order linear hyperbolic problems. This method combines the benefit of the discontinuous Galerkin method and the streamline diffusion finite element method. Two fully discrete DFDSD schemes (Euler DFDSD and Crank–Nicolson (CN) DFDSD) are constructed by making use of the difference discrete method for time variables and the discontinuous streamline diffusion method for space variables. The stability and optimal L² norm error estimates are established for the constructed schemes. This method makes contributions to the discontinuous methods. Finally, a numerical example is provided to show the benefit of high efficiency and simple implementation of the schemes.     

Non-oscillation of solutions of difference equations of third order

In this paper, sufficient conditions in terms of coefficient functions are obtained for non-oscillation of all solutions of a class of linear homogeneous third order difference equations of the form

     

The finite difference method for first order impulsive partial differential-functional equations

We consider initial boundary value problems for first order impulsive partial differential-functional equations. We give sufficient conditions for the convergence of a general class of one step difference methods. We assume that given functions satisfy the non-linear estimates of the Perron type with respect to the functional argument. The proof of stability is based on a theorem on difference functional inequalities generated by an impulsive differential-functional problem. It is an essential assumption in our consideration that given functions satisfy the Volterra condition. We give a numerical example.     

Higher order finite strip analysis for curved bridge decks

The development of a higher order finite strip method for improved accuracy and its application to orthotropic curved bridge decks are discussed. A quintic polynomial in the radial direction is employed along with a basic function series in the angular direction which satisfy a priori the boundary conditions along the radial edges. Thus a two-dimensional plate bending problem is reduced to a one-dimensional one. As a result, both the size and bandwidth of the global stiffness matrix are greatly reduced. The method is easy to program, requires a minimum input data and small computer storage. In order to estimate the reliability of the present formulation, three examples of curved plates are solved and the results are compared with the existing solutions.     

Higher order meshless schemes applied to the finite element method in elliptic problems

This paper presents selected approximation techniques, typical for the meshless finite difference method (MFDM), although applied to the finite element method (FEM). Finite elements with standard or hierarchical shape functions are coupled with higher order meshless schemes, based upon the correction terms of a simple difference operator. Those terms consist of higher order derivatives, which are evaluated by means of the appropriate formulas composition as well as a numerical solution, which corresponds to the primary interpolation order, assigned to element shape functions. Correction terms modify the right-hand sides of algebraic FE equations only, yielding an iterative procedure. Therefore, neither re-generation of the stiffness matrix nor introduction of any additional nodes and/or degrees of freedom is required. Such improved FE-MFD solution approach allows for the optimal application of advantages of both methods, for instance, a high accuracy of the nodal FE solution and a derivatives’ super-convergence phenomenon at arbitrary domain points, typical for the meshless FDM. Existing and proposed higher order techniques, applied in the FEM, are compared with each other in terms of the solution accuracy, algorithm efficiency and computational complexity.In order to examine the considered algorithms, numerical results of several two-dimensional benchmark elliptic problems are presented. Both the accuracy of a solution and the solution’s derivatives as well as their convergence rates, evaluated on irregular and structured meshes as well as arbitrarily irregular adaptive clouds of nodes, are taken into account.     

High order finite difference methods for unsteady incompressible flows in multi-connected domains

Using the vorticity and stream function variables is an effective way to compute 2-D incompressible flow due to the facts that the incompressibility constraint for the velocity is automatically satisfied, the pressure variable is eliminated, and high order schemes can be efficiently implemented. However, a difficulty arises in a multi-connected computational domain in determining the constants for the stream function on the boundary of the “holes”. This is an especially challenging task for the calculation of unsteady flows, since these constants vary with time to reflect the total fluxes of the flow in each sub-channel. In this paper, we propose an efficient method in a finite difference setting to solve this problem and present some numerical experiments, including an accuracy check of a Taylor vortex-type flow, flow past a non-symmetric square, and flow in a heat exchanger.     

On the strong stability of finite difference schemes for hyperbolic systems in two space dimensions

We study the stability of some finite difference schemes for symmetric hyperbolic systems in two space dimensions. For the so-called upwind scheme and the Lax–Wendroff scheme with a stabilizer, we show that stability is equivalent to strong stability, meaning that both schemes are either unstable or $\ell ^2$ -decreasing. These results improve on a series of partial results on strong stability. We also show that, for the Lax–Wendroff scheme without stabilizer, strong stability may not occur no matter how small the CFL parameters are chosen. This partially invalidates some of Turkel’s conjectures in Turkel (16(2):109–129, 1977).     

A fourth order finite difference method for waveguides with curved perfectly conducting boundaries

A novel high order finite difference method is introduced for optical waveguides with smoothly curved perfectly electric conducting (PEC) boundaries. The proposed method shares some similarities with our previous matched interface and boundary (MIB) methods developed for treating dielectric interfaces of optical waveguides, such as the use of a simple Cartesian grid, the standard finite difference schemes, and fictitious values. However, the PEC boundary conditions have a physical nature quite different from that of the jump conditions at the dielectric interfaces, i.e., all six electric and magnetic field components are prescribed in the jump conditions, while only three of them are known at the PEC walls. Consequently, the previously developed MIB methods are not applicable to deal with the perfectly conducting boundaries. To overcome this difficulty, a novel ray-casting fictitious domain method is constructed to enforce the PEC conditions along the normal direction. Such a boundary implementation couples the transverse magnetic field components so that the resulting ray-casting MIB method is a full vectorial approach for the modal analysis of optical waveguides. The new MIB method is validated by considering both homogeneous and inhomogeneous waveguides. Numerical results confirm the designed fourth order of accuracy.     

Higher order dual Lagrange multiplier spaces¶for mortar finite element discretizations

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements. In particular, we focus on dual Lagrange multiplier spaces. These non-standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We construct locally supported and continuous dual basis functions for quadratic finite elements starting from the discontinuous quadratic dual basis functions for the Lagrange multiplier space. In particular, we compare different dual Lagrange multiplier spaces and piecewise linear and quadratic finite elements. The optimality of the associated mortar method is shown. Numerical results illustrate the performance of our approach. Received: July 2002 / Accepted: November 2002     

Upwind finite volume solution of sensitivity equations for hyperbolic systems of conservation laws with discontinuous solutions

An upwind, finite volume method is proposed for the direct solution of one-dimensional, hyperbolic systems-based sensitivity equations. Sensitivity equations for hyperbolic systems of conservation laws require a specific treatment of discontinuities, across which Dirac source terms appear, thus leading to modify the classical jump relationships. The modified jump relationships are used to derive an extension of the HLL and HLLC Riemann solvers for the solution of one-dimensional, hyperbolic systems-based sensitivity equations. The solver is developed for 3 × 3 systems where the central wave is a contact wave. A specific treatment is needed to preserve the invariance property of the third component of the sensitivity vector along the contact wave. The proposed solver is applied to the one-dimensional, Saint Venant equations with passive, scalar advection and tested successfully against analytical solutions including the influence of topography-induced source terms in both the flow and sensitivity equations.     

Asymptotic and oscillatory behavior of solutions of general nonlinear difference equations of second order

The asymptotic and oscillatory behavior of solutions of some general second-order nonlinear difference equations of the form

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=0 nZ,

is studied. Oscillation criteria for their solutions, when “p_n” is of constant sign, are established. Results are also presented for the damped-forced equation

δ(a_nh(y_n+1)δy_n)+p_nδy_n+q_n+1f(y_σ(n+1))=e_n nZ.

Examples are inserted in the text for illustrative purposes.     

Regularity results for discrete solutions of second order elliptic problems in the finite element method

We discuss a discrete version of the De Giorgi-Nash-Moser regularity theory for solutions of elliptic second-oder equations. Working under a fellowship of Consejo Nacional de Investigationes Cientificas y Tecnicas, Argentina Partially supported by National Science Foundation Grant MCS7915171