Stability Analysis Of Numerical Methods Research Articles

Nonlinear systems of parabolic IBVP: A stable super-supraconvergent fully discrete piecewise linear FEM

The main purpose of this paper is the design of discretizations for second order nonlinear parabolic initial boundary value problems which are stable and present second order of convergence with respect to H1-discrete norms. The convergence results are established assuming that the solutions are in H3. The stability analysis of numerical methods around a numerical solution requires the uniform boundness of such solution. Although such bounds are usually taken as an assumption, in this paper these will be deduced as a corollary of suitable error estimates. As the methods can be simultaneously seen as piecewise linear finite element methods and finite difference methods, the convergence results can be seen simultaneously as superconvergence results and supraconvergence results. Numerical results illustrating the sharpness of the smoothness assumptions and an application to simulation of the solar magnetic field in the umbra (the central zone of a sunspot) are also included.

A generalized von Neumann analysis for multi-level schemes: Stability and spectral accuracy

The so-called von Neumann analysis is a well-established approach used for stability analysis of numerical methods. The crux of this analysis is to bound the amplification factor by unity to ensure stability. However, an implicit but commonly unverified assumption in this approach is that the amplification factor does not vary with time, which as we show here is not always true for multi-level schemes. We propose a generalized von Neumann analysis wherein we take into account the temporal variation of the amplification factor and thus overcome the limitations of the standard analysis. We express this time-varying amplification factor as a continued fraction and obtain exact conditions for the applicability of the standard von Neumann approach. We define stability in terms of product of the amplification factor at all times that allows the instantaneous amplification to be larger than unity. This is indeed observed in simulations though the scheme remains stable which makes it then, unexplainable with the standard von Neumann analysis. We use the proposed generalized analysis and stability definition to assess the stability of asynchrony-tolerant schemes with periodic coefficients. The degrading effect of temporal scheme on the spectral accuracy of spatial schemes at large CFL values is also discussed.

Analysis of Perturbed Volterra Integral Equations on Time Scales

This paper describes the effect of perturbation of the kernel on the solutions of linear Volterra integral equations on time scales and proposes a new perspective for the stability analysis of numerical methods.

On the asymptotic stability of Volterra functional equations with vanishing delays

We analyze the asymptotic stability of solutions of linear Volterra integral equations with general continuous convolution kernels and vanishing delays. The analysis is based on an extension of the variation-of-parameter formula for non-delay Volterra integral equations and on energy function techniques. The delay integral equations studied in this paper will be of interest in the (still open) stability analysis of numerical methods (e.g. collocation and Runge-Kutta-type methods) for Volterra integral equations with vanishing delays.

Non-normal drift structures and linear stability analysis of numerical methods for systems of stochastic differential equations

We investigate mean-square asymptotic stability of equilibria in linear systems of stochastic differential equations with non-normal drift coefficients, with particular emphasis on the role of interactions between the drift and diffusion structures that act along, orthogonally to, and laterally to the flow. Hence we construct test systems with non-normal drift coefficients and characteristic diffusion structures for the purposes of a linear stability analysis of the θ-Maruyama method. Next we discretise these test systems and examine the mean-square asymptotic stability of equilibria of the resulting systems of stochastic difference equations. Finally we indicate how this approach may help to shed light on numerical discretisations of stochastic partial differential equations with multiplicative space–time perturbations.

Delay-dependent stability analysis of numerical methods for stochastic delay differential equations

This paper is concerned with the numerical solution of stochastic delay differential equations. The focus is on the delay-dependent stability of numerical methods for a linear scalar test equation with real coefficients. By using the so-called root locus technique, the full asymptotic stability region in mean square of stochastic theta methods is obtained, which is characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficients as well as time stepsize and method parameter theta. Then, this condition is compared with the analytical stability condition. It is proved that the Backward Euler method completely preserves the asymptotic mean square stability of the underlying system and the Euler–Maruyama method preserves the instability of the system. Our investigation also shows that not all theta methods with θ≥12 preserve this delay-dependent stability. Some numerical examples are presented to confirm the theoretical results.

A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems

We are concerned with a linear mean-square stability analysis of numerical methods applied to systems of stochastic differential equations (SDEs) and, in particular, consider the θ-Maruyama and the θ-Milstein method in this context. We propose an approach, based on the vectorisation of matrices and the Kronecker product, that allows us to deal efficiently with the matrix expressions arising in this analysis and that provides the explicit structure of the stability matrices in the general case of linear systems of SDEs. For a set of simple test SDE systems, incorporating different noise structures but only a few parameters, we apply the general results and provide visual and numerical comparisons of the stability properties of the two methods.

THE NUMERICAL STABILITY OF STOCHASTIC ORDINARY DIFFERENTIAL EQUATIONS WITH ADDITIVE NOISE

An asymptotic stability analysis of numerical methods used for simulating stochastic differential equations with additive noise is presented. The initial part of the paper is intended to provide a clear definition and discussion of stability concepts for additive noise equation derived from the principles of stability analysis based on the theory of random dynamical systems. The numerical stability analysis presented in the second part of the paper is based on the semi-linear test equation dX(t) = (AX(t) + f(X(t))) dt + σ dW(t), the drift of which satisfies a contractive one-sided Lipschitz condition, such that the test equation allows for a pathwise stable stationary solution. The θ-Maruyama method as well as linear implicit and two exponential Euler schemes are analysed for this class of test equations in terms of the existence of a pathwise stable stationary solution. The latter methods are specifically developed for semi-linear problems as they arise from spatial approximations of stochastic partial differential equations.

Towards a Systematic Linear Stability Analysis of Numerical Methods for Systems of Stochastic Differential Equations

We develop two classes of test equations for the linear stability analysis of numerical methods applied to systems of stochastic ordinary differential equations of Itô type (SODEs). Motivated by the theory of stochastic stabilization and destabilization, these test equations capture certain fundamental effects of stochastic perturbation in systems of SODEs, while remaining amenable to analysis before and after discretization. We then carry out a linear stability analysis of the $\theta$-Maruyama method applied to these test equations, investigating mean-square and almost sure asymptotic stability of the test equilibria. We discuss the implications of our work for the notion of A-stability of the $\theta$-Maruyama method and use numerical simulation to suggest extensions of our results to test systems with nonnormal drift coefficients. (A corrected version of this paper has been appended to the originally posted pdf.)

Stability of θ-methods for advanced differential equations with piecewise continuous arguments

This paper deals with the stability analysis of numerical methods for the solution of advanced differential equations with piecewise continuous arguments. We focus on the behaviour of the one-leg θ-method and the linear θ-method in the solution of the equation x′( t) = ax( t + a 0 x([ t]) + a 1 x([ t+1]), with real a, a 0, a 1 and [·] designates the greatest-integer function. The stability regions of two θ-methods are determined. The conditions under which the analytic stability region is contained in the numerical stability region are obtained and some numerical experiments are given.

Stability analysis of numerical methods for linear neutral Volterra delay-integro-differential system

This paper is concerned with the stability of numerical methods for linear neutral Volterra delay-integro-differential system. A sufficient condition such that the system is asymptotically stable is derived. Furthermore, it is proved that every linear θ-method with θ ∈ (1/2, 1] and A-stable BDF method preserve the delay-independent stability of its exact solutions. Finally, the numerical experiments are given to demonstrate the conclusions.

Analysis of error growth via stability regions in numerical initial value problems

This paper concerns the stability analysis of numerical methods for solving time dependent ordinary and partial differential equations. In the literature stability estimates for such methods were derived, under a condition which can be viewed as a transplantation of the Kreiss resolvent condition (from the unit disk to the stability region S of the numerical method). These estimates tell us that errors in the numerical time stepping process cannot grow faster than linearly with min{s, n} .H eren denotes the number of time steps, and s stands for the order of the (spatial discretization) matrices involved. In this paper we address the natural question of whether the above stability estimates can be improved so as to imply an error growth at a slower rate than min{s, n} (when n →∞ , s →∞ ). Our results concerning this question are as follows: (a) for all (practical) Runge–Kutta and other one-step formulas, we show that the estimates from the literature are sharp in that error growth at the rate min{s, n} can actually occur, (b) for linear multistep formulas we find that, rather surprisingly, some of the stability estimates can substantially be improved and extended, whereas others are sharp. The results proved in this paper are also relevant to (suitably scaled spatial discretization) matrices whose e-pseudo-eigenvalues lie at a distance not exceeding Ke from the stability region S of the time stepping method, for all e> 0 and fixed constant K. AMS subject classification (2000): Primary: 65L20; Secondary: 65L05, 65L06, 65M12, 65M20.

Stability analysis of numerical methods for systems of functional-differential and functional equations

This paper is concerned with the numerical solution of functional-differential and functional equations which include functional-differential equations of neutral type as special cases. The adaptation of linear multistep methods, one-leg methods, and Runge-Kutta methods is considered. The emphasis is on the linear stability of numerical methods. It is proved that A-stable methods can inherit the asymptotic stability of underlying linear systems. Some general results of stability on explicit and implicit methods are also given.

Resolvent conditions and bounds on the powers of matrices, with relevance to numerical stability of initial value problems

We deal with the problem of establishing upper bounds for the norm of the nth power of square matrices. This problem is of central importance in the stability analysis of numerical methods for solving (linear) initial value problems for ordinary, partial or delay differential equations. A review is presented of upper bounds which were obtained in the literature under the resolvent condition occurring in the Kreiss matrix theorem, as well as under variants of that condition. Moreover, we prove new bounds, under resolvent conditions which generalize some of the reviewed ones. The paper concludes by applying one of the new upper bounds in a stability analysis of the trapezoidal rule for delay differential equations.

Mean-Square and Asymptotic Stability of the Stochastic Theta Method

Stability analysis of numerical methods for ordinary differential equations (ODEs) is motivated by the question "for what choices of stepsize does the numerical method reproduce the characteristics of the test equation?" We study a linear test equation with a multiplicative noise term, and consider mean-square and asymptotic stability of a stochastic version of the theta method. We extend some mean-square stability results in [Saito and Mitsui, SIAM. J. Numer. Anal., 33 (1996), pp. 2254--2267]. In particular, we show that an extension of the deterministic A-stability property holds. We also plot mean-square stability regions for the case where the test equation has real parameters. For asymptotic stability, we show that the issue reduces to finding the expected value of a parametrized random variable. We combine analytical and numerical techniques to get insights into the stability properties. For a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region. This allows us to prove a number of results. The technique introduced is widely applicable, and we use it to show that a fully implicit method suggested by [Kloeden and Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, 1992] has an asymptotic stability extension of the deterministic A-stability property. We also use the approach to explain some numerical results reported in [Milstein, Platen, and Schurz, SIAM J. Numer. Anal., 35 (1998), pp. 1010--1019.]

Backward Error Analysis for Lie-Group Methods

Backward error analysis has proven to be very useful in stability analysis of numerical methods for ordinary differential equations. However the analysis has so far been undertaken in the Euclidean space or closed subsets thereof. In this paper we study differential equations on manifolds. We prove a backward error analysis result for intrinsic numerical methods. Especially we are interested in Lie-group methods. If the Lie algebra is nilpotent a global stability analysis can be done in the Lie algebra. In the general case we must work on the nonlinear Lie group. In order to show that there is a perturbed differential equation on the Lie group with a solution that is exponentially close to the numerical integrator after several steps, we prove a generalised version of Alekseev-Gr: obner's theorem. A major motivation for this result is that it implies many stability properties of Lie-group methods.

Electrical Impedance Tomography

t. This paper surveys some of the work our group has done in electrical impedance tomography.

Resolvent conditions for discretizations of diffusion-convection-reaction equations in several space dimensions

Resolvent conditions are useful in the stability analysis of numerical methods for the solution of linear initial value problems. Under such conditions one can prove stability estimates in terms of the number of time steps n and the dimension s of the space. This paper addresses the question of whether those resolvent conditions are fulfilled if the linear initial value problems under consideration arise from the semi-discretization of a diffusion-convection-reaction equation in several space dimensions. In this paper some relevant results from the literature are reviewed, some of these results are extended, and some new results are proved.

A Note on the Order of Contact between Sets in the Complex Plane

This paper deals with the order of contact between arbitrary setsXandYlying in the complex plane C. Let ϕ be a rational function, and ξ0∈C. Suppose, at all pointsxi∈C∪{∞} with ϕ(xi)=ξ0, the setsX,Yhave contact of a given orderpi. In the stability analysis of numerical methods for solving differential equations, the problem arises to determine the order of contact, sayp, between ϕ(X) and ϕ(Y) at ξ0. In this paper a theorem is given according to whichp=maxipi/li, whereli≥1 denotes the order of a certain nonvanishing derivative related to ϕ andxi. The theorem is valid under the assumption that ϕ−1(ϕ(X))⊂cl(X) or ϕ−1(ϕ(Y))⊂cl(Y). This assumption is satisfied in the stability analysis just mentioned, so that the theorem settles the problem arising in that context. The paper also deals with various natural questions related to the specific concept of order (of contact) used in the above mentioned theorem. Some of these questions are important in view of applications.

Stability estimates for families of matrices of nonuniformly bounded order

This paper addresses the problem of establishing upper bounds for the norm of the nth power of square matrices. The focus is on upper bounds that are valid for families of matrices of finite, but nonuniformly bounded, order. Such upper bounds are relevant to the stability analysis of numerical methods for solving differential equations. In the famous Kreiss matrix theorem a condition, on the resolvent of matrices, occurs which implies that the norm of the nth power does not grow faster than linearly with n. In this paper a slightly stronger version of this resolvent condition is studied, which is often satisfied in cases of practical interest. We prove that this version implies growth at a rate that is essentially lower than in the case of the classical Kreiss resolvent condition. We consider also a condition, on the iterated resolvent of matrices, which is related to one of the assumptions occurring in the Hille-Yosida theorem. This condition is known to imply growth at a rate C(n 1 2 ) . We prove that, when a slightly stronger version of the iterated resolvent condition is in force, there is growth at a rate that is at most C( n p ), with 0 ≤ p < 1 2 .