The stability of boundary conditions for an angled-derivative difference scheme

Tidal forcing of the shallow water equations is typical of a class of problems where an approximate equilibrium solution is sought by long time integration of a differential equation system. A combination of the angled-derivative scheme with a staggered leap-frog scheme is sometimes used to discretise this problem. It is shown here why great care then needs to be taken with the boundary conditions to ensure that spurious solution modes do not lead to numerical instabilities. Various techniques are employed to analyse two simple model problems and display instabilities met in practical computations; these are then used to deduce a set of stable boundary conditions.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

An adaptive Euler-Maruyama scheme for SDEs: convergence and stability

J. C. Mattingly ^{The understanding of adaptive algorithms for stochastic differentialequations (SDEs) is an open area, where many issues relatedto both convergence and stability (long-time behaviour) of algorithmsare unresolved. This paper considers a very simple adaptivealgorithm, based on controlling only the drift component ofa time step. Both convergence and stability are studied. Theprimary issue in the convergence analysis is that the adaptivemethod does not necessarily drive the time steps to zero withthe user-input tolerance. This possibility must be quantifiedand shown to have low probability. The primary issue in thestability analysis is ergodicity. It is assumed that the noiseis nondegenerate, so that the diffusion process is elliptic,and the drift is assumed to satisfy a coercivity condition.The SDE is then geometrically ergodic (averages converge tostatistical equilibrium exponentially quickly). If the driftis not linearly bounded, then explicit fixed time step approximations,such as the Euler–Maruyama scheme, may fail to be ergodic.In this work, it is shown that the simple adaptive time-steppingstrategy cures this problem. In addition to proving ergodicity,an exponential moment bound is also proved, generalizing a resultknown to hold for the SDE itself.}     

Vacuum States and Equidistribution of the Random Sequence for Glimm's Scheme (continuation)

This is a continuation of paper [1]. The difference between this paper and paper [1] is that the initial functions considered here are step functions and those considered in [1] are. Lipschitz continuous. Since there are centered rarefaction waves here, more delicate techniques are needed. It may be a necessary step in solving p-System with general initial functions by Glimm's scheme. Notice that this paper can not be deduced from [1].     

A general fuzzy control scheme for nonlinear processes with stability analysis

In this paper we develop a general fuzzy control scheme for nonlinear processes. Assuming little knowledge about the dynamics of the controlled process, the proposed scheme starts by probing the process at different points in its operating region to generate a fuzzy quantisation. A simple local controller is then designed at each fuzzy locality. A fuzzy inference mechanism then links up tje local controllers to form a global controller which can be further refined by the learning algorithm. By employing a newly developed structure-adaptive fuzzy modelling scheme, the appropriate fuzzy rule-base for the inference mechanism can be extracted stably and efficiently. The conditions for the stability of the global controller are rigourously established. Simulation results are presented to illustrate the effectiveness of the scheme.     

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.     

A second order scheme for the navier-stokes equations: stability and convergence

In this paper, a fully discretized projection method is introduced. It contains a parameter operator. Depending on this operator, we can obtain a first-order scheme, which is appropriate for theoretical analysis, and a second-order scheme, which is more suitable for actual computations. In this method, the boundary conditions of the intermediate velocity field and pressure are not needed. We give the proof of the stability and convergence for the first-order case. For the higher order cases, the proof were different, and we will present it elsewhere.

In a forthcoming article [7], we apply this scheme to the driven-cavity problem and compare it with other schemes     

Numerical stability analysis of an acceleration scheme for step size constrained time integrators

Time integration schemes with a fixed time step, much smaller than the dominant slow time scales of the dynamics of the system, arise in the context of stiff ordinary differential equations or in multiscale computations, where a microscopic time-stepper is used to compute macroscopic behaviour. We discuss a method to accelerate such a time integrator by using extrapolation. This method extends the scheme developed by Sommeijer [Increasing the real stability boundary of explicit methods, Comput. Math. Appl. 19(6) (1990) 37–49], and uses similar ideas as the projective integration method. We analyse the stability properties of the method, and we illustrate its performance for a convection–diffusion problem.     

On the global stability of a temporal discretization scheme for the Navier-Stokes equations

The time discretization by a linear backward Euler scheme forthe non-stationary viscous incompressible Navier–Stokesequations with a non-zero external force in a bounded 2D domainwith no-slip boundary condition or periodic boundary conditionis studied. Improved global stability results are obtained. The boundedness of the solution sequence in V and D(A) normsuniform with respect to &t for t [0, ) is proved. A similarresult in the V norm was previously obtained by (Geveci, 1989Math. Comp., 53, 43–53) for the non-forced system. A differentapproach is used here. As a corollary, the global attractorfor the approximation scheme is proved to exist, which is boundedin both V and D(A) spaces, thus compact in both H and V spaces.Applying the same techniques developed here, we are able toimprove the main result of (Hill and Süli 2000 IMA J. Numer.Anal., 20, 633–667) by showing that besides the existenceof a global attractor, the whole solution sequence is uniformlybounded in V as well, which is of significance from the pointof view of computing. As a corollary of local convergence results,upper semi-continuity of the attractor with respect to the numericalperturbation induced by the linear scheme is also establishedin both H and V spaces. Finally, some preliminary estimates,which are to our knowledge the first of their kind, on the dimensionsof the attractors in H and V spaces are also obtained.     

One-sided stability and convergence of the Nessyahu–Tadmor scheme

Non-oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws. The Nessyahu–Tadmor (NT) scheme is an example of a second order scheme that is both robust and simple. In this paper, we prove a new stability property of the NT scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law. This property is similar to the One-sided Lipschitz condition for first order schemes. Using this new stability, we derive the convergence of the NT scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method. We also derive an error estimate for monotone initial data.     

Existence and stability of stationary profiles of the lw scheme

In this paper we study the behavior of difference schemes approximating solutions with shocks of scalar conservation laws When a difference scheme introduces artificial numerical diffusion, for example the Lax-Friedrichs scheme, we experience smearing of the shocks, whereas when a scheme introduces numerical dispersion, for example the Lax-Wendroff scheme, we experience oscillations which decay exponentially fast on both sides of the shock. In his dissertation. Gray Jennings studied approximation by monotone schemes. These contain artificial viscosity and are first-order accurate; they are known to be contractive in the sense of any l^p norm. Jennings showed existence and l¹ stability of traveling discrete smeared shocks for such schemes. Here we study similar questions for the Lax-Wendroff scheme without artificial viscosity; this is a nonmonotone, second-order accurate scheme. We prove existence of a one-parameter family of stationary profiles. We also prove stability of these profiles for small perturbations in the sense of a suitably weighted l² norm. The proof relies on studying the linearized Lax-Wendroff scheme.     

On the stability of an explicit difference scheme for hyperbolic equations with nonlocal boundary conditions

We consider the stability of an explicit finite-difference scheme for a linear hyperbolic equation with nonlocal integral boundary conditions. By studying the spectrum of the transition matrix of the explicit three-layer difference scheme, we obtain a sufficient condition for stability in a special norm.     

A note on a discretization scheme for Volterra integro-differential equations that preserves stability and boundedness

We extend the validity of some recent results on discretization of scalar Volterra differential equations.     

Convergence and stability of the finite difference scheme for nonlinear parabolic systems with time delay

In this article, a class of nonlinear evolution equations – reaction–diffusion equations with time delay – is studied. By combining the domain decomposition technique and the finite difference method, the results for the existence, convergence and the stability of the numerical solution are obtained in the case of subdomain overlap and when the time-space is completely discretized. This revised version was published online in June 2006 with corrections to the Cover Date.     

Discretization scheme in Volterra integro-differential equations that preserves stability and boundedness

A nonstandard discretization scheme is applied to continuous Volterra integro-differential equations. We will show that under our discretization scheme the stability of the zero solution of the continuous dynamical system is preserved. Also, under the same discretization, using a combination of Lyapunov functionals, Laplace transforms and z-transforms, we show that the boundedness of solutions of the continuous dynamical system is preserved.     

On the stability of an implicit spline collocation difference scheme for linear partial differential algebraic equations

A boundary value problem for linear partial differential algebraic systems of equations with multiple characteristic curves is examined. It is assumed that the pencil of matrix functions associated with this system is smoothly equivalent to a special canonic form. The spline collocation is used to construct for this problem a difference scheme of an arbitrary approximation order with respect to each independent variable. Sufficient conditions are found for this scheme to be absolutely stable.     

Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids

A diffusion driven model for hepatitis B virus (HBV) infection, taking into account the spatial mobility of both the HBV and the HBV DNA-containing capsids is presented. The global stability for the continuous model is discussed in terms of the basic reproduction number. The analysis is further carried out on a discretized version of the model. Since the standard finite difference (SFD) approximation could potentially lead to numerical instability, it has to be restricted or eliminated through dynamic consistency. The latter is accomplished by using a non-standard finite difference (NSFD) scheme and the global stability properties of the discretized model are studied. The results are numerically illustrated for the dynamics and stability of the various populations in addition to demonstrating the advantages of the usage of NSFD method over the SFD scheme.