Stability Analysis Of Difference Schemes Research Articles

Computational Experiment for Stability Analysis of Difference Schemes with Nonlocal Conditions

Based on an example, we describe how outcomes of computational experiment can be employed for study of stability of numerical algorithm, provided that related theoretical propositions are not proven yet. More precisely, we propose a systematic and generalized methodology, how to investigate the influence of the weight functions α(x) and β(x), present in the integral boundary conditions, on the stability of difference schemes, for some class of parabolic equations. The ground of the methodology is the investigation of the spectrum of a matrix, defining the transition to the upper layer of the difference scheme. Spectral structure of this matrix is analysed by both analytic method and computational experiment.

On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems

We deal with the stability analysis of difference schemes for a one-dimensional parabolic equation subject to integral conditions. It is based on the spectral structure of the transition matrix of a difference scheme. The stability domain is defined by using the hyperbola which is the locus of points where the transition matrix has trivial eigenvalues. The stability conditions obtained are much more general compared with those known in the literature. We analyze three separate cases of nonlocal integral conditions and solve an example illustrating the efficiency of the technique.

Parallel implementation of stability analysis of difference schemes with Mathematica

We consider a parallel algorithm for investigating the stability of the schemes of the finite-difference and finite-volume methods that approximate the two-dimensional Euler equations of compressible fluid on a curvilinear grid. The algorithm is implemented with the aid of the computer algebra system Mathematica 3.0. We apply a two-level parallelization process. At the first level, the symbolic computation of the amplification matrix is parallelized by a parallel computation of the matrix rows on different processors. At the second level, the values of the coordinates of points of the stability-region boundary are computed numerically. For the communication between the workstations, we apply a special program, LaunchSlave, which uses the MathLink communication protocol. Examples of application of the proposed parallel symbolic/numerical algorithm are presented. Bibliography: 15 titles.

Application of Computer Algebra Systems for Stability Analysis of Difference Schemes on Curvilinear Grids

The paper deals with problems arising in the application of the computer algebra systems for the symbolic–numeric stability analysis of difference schemes and schemes of the finite-volume method approximating the two-dimensional Euler equations for compressible fluid flows on curvilinear spatial grids. We carry out a detailed comparison of the REDUCE 3.6 and Mathematica(Versions 2.2 and 3.0) from the point of view of their applicability to the solution of the above problems. We draw a conclusion that a preference should be given for Mathematica from the viewpoint of the execution of symbolic–numeric computations. We also describe in detail our new symbolic–numeric algorithm for stability investigation, which was implemented with the aid of Mathematica. The proposed method enables us to reduce the needed computer storage at the symbolic stages by a factor of about 20 as compared with the previous algorithms. A feature of the numerical stages is the use of the arithmetic of rational numbers, which enables us to avoid the accumulation of the roundoff errors. We present the examples of the application of the proposed symbolic–numeric method for stability analysis of very complex schemes of the finite-volume method on curvilinear grids, which are widely used in computational fluid dynamics.

Applying Quantifier Elimination to Stability Analysis of Difference Schemes

Stability analysis is an important tool for constructing time-stepping finite difference schemes for partial differential equations. This paper describes how von Neumann stability analysis can be reduced to a quantifier elimination problem over the reals. We report our experience in analysing some difference schemes by using a quantifier package based on the partial cylindrical algebraic decomposition algorithm.

Stability analysis of difference schemes by catastrophe theory methods

When analyzing the stability of difference schemes by the Fourier method one usually investigates the location of zeros some algebraic equation with respect to a unit circle in a complex plane. We propose to investigate instead of this equation the behaviour of a family of curves described by a different algebraic equation whose coefficients are computed on the basis of the coefficients of the above equation and depend on the parameters, the number of which is equal to the number of space variables in the original Cauchy problem for a partial differential equation with constant coefficients. One of the curves of this family describes the behaviour of the stability domain boundary of a difference scheme. It is shown by examples of two well-known difference schemes how the mathematical apparatus of catastrophe theory can be used effectively for the detection of the domains of instability and of stability of a difference scheme. Based on the technique we propose a method for the numerical determination of the stability domain boundary of a difference scheme.

Stability Analysis of Difference Schemes for Variable Coefficient Schrödinger Type Equations

We consider the stability of difference schemes for the solution of the initial boundary value problem for the equation \[ u_1 = \left( {A(x,t)u_x } \right)_x + B(x,t)u + C(x,t)u + f(x,t)\], where u, A, B, C and f are complex valued functions. Using energy methods, we establish the stability of a general two level scheme which includes Euler’s method, Crank–Nicolson’s method and the backward Euler method. If the coefficient $A(x,t)$ is purely imaginary, the explicit Euler method is unconditionally unstable. For this case, we propose a new scheme with appropriately chosen artificial dissipation, which we prove to be conditionally stable.

The stability analysis of difference schemes by numerical solution of the generalised Routh-Hurwltz problem

A numerical method is presented for the stability investigation of difference schemes approximating the systems of partial differential equations with constant coefficients. The method enables one to determine with a sufficiently high accuracy the locus of points of the boundary of the stability domain of a difference scheme on the basis of the necessary stability criteria.

The stability of explicit difference schemes for solving the problem of interaction between a compressible fluid and an elastic shell

A mathematical model for describing the interaction between a compressible fluid and an elastic shell is formulated as an initial boundary value problem. The partial differential equations of the model are discretized both in time and space by a finite-difference method. The stability of the resulting explicit difference schemes is analyzed by Kreiss' theory for the stability analysis of difference schemes in initial boundary value problems. It is shown that the stability properties of the schemes for the interaction problem may be influenced by the type of discretization in space used for the contact condition on the interface between the fluid and the shell and also by the approximation of the hydrodynamic pressure on the surface of the shell. A simple sufficient condition is found that will ensure the best possible stability properties of the schemes. Several of these, which are of practical interest, are analyzed.