On the use of stability regions in the numerical analysis of initial value problems

Lenferink, H. W. J.; Spijker, M. N.

doi:10.1090/s0025-5718-1991-1079026-6

Cited by 34 publications

(32 citation statements)

References 21 publications

(19 reference statements)

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“…Although we are able to guarantee the weaker stability estimate (6.1) using numerical range arguments [8], our numerical experiments indicate that the algebraic growth, k 1/2 , does not occur in practice. If we scan across the rows of Table 6.1 we see that as k → ∞, B k n ∞ decays.…”

Section: Non-legitimacy Of the Pde Boundary Conditionmentioning

confidence: 77%

“…Using a Dirichlet node, the convex hull of the Gerschgorin disks of A lies in the right half plane. From [8] this allows us to guarantee algebraic stability in the sense that:…”

Section: Non-legitimacy Of the Pde Boundary Conditionmentioning

confidence: 99%

“…In our case the matrix is non-symmetric and non-normal, and for non-normal matrices, counterexamples can be given where, even if the eigenvalues are less than one in magnitude, instability results as the dimension of the matrix becomes large (see [11,7,9,17]). For some values of the market parameters we are able to show that sufficient conditions for stability are satisfied using numerical range arguments [7,8,16,3]. In other cases, these arguments cannot be applied and we follow [2] and *…”

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confidence: 99%

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Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Windcliff¹,

Forsyth²,

Vetzal³

2004

JCF

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Abstract. The linear asymptotic boundary condition, i.e. assuming that the second derivative of the value of the derivative security vanishes as the asset price becomes large, is commonly used in practice. To our knowledge, there have been no rigorous studies of the stability of these methods, despite the fact that the discrete matrix equations obtained using this boundary condition loses diagonal dominance for large timesteps. In this paper, we demonstrate that the discrete equations obtained using this boundary condition satisfy necessary conditions for stability for a finite difference discretization. Computational experiments also show that this boundary condition satisfies sufficient conditions for stability as well. [19,18,10] have recommended a linear asymptotic boundary condition (that the second derivative of the option value with respect to the underlying asset value be zero) as the asset price becomes large. Although this boundary specification is often applied, to our knowledge there have been no rigorous studies of the stability of this technique. Looking at the form of the discrete matrix equations obtained using this boundary condition, the resulting discrete equations lose diagonal dominance for large timesteps and the usual arguments cannot be applied to guarantee unconditional stability.In order to determine the ranges of parameters (risk-free rate, volatility, etc.) for which this asymptotic boundary condition could cause instability, we derive necessary conditions for the stability of the discrete equations based on the spectrum of the matrix representing the spatial discretization. Somewhat surprisingly, we find that a finite difference (FD) discretization always satisfies these necessary conditions for stability, despite the fact that the matrix equations are not unconditionally diagonally dominant.The eigenvalues can be used to determine necessary conditions for stability but are known to be unreliable for determining sufficient conditions for stability. For finite dimensional problems analysis of the spectrum can lead to sufficient conditions but in the PDE context, the size of the matrix becomes unbounded as the grid is refined. In our case the matrix is non-symmetric and non-normal, and for non-normal matrices, counterexamples can be given where, even if the eigenvalues are less than one in magnitude, instability results as the dimension of the matrix becomes large (see [11,7,9,17]). For some values of the market parameters we are able to show that sufficient conditions for stability are satisfied using numerical range arguments [7,8,16,3]. In other cases, these arguments cannot be applied and we follow [2] and *

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Section: Non-legitimacy Of the Pde Boundary Conditionmentioning

confidence: 77%

“…Using a Dirichlet node, the convex hull of the Gerschgorin disks of A lies in the right half plane. From [8] this allows us to guarantee algebraic stability in the sense that:…”

Section: Non-legitimacy Of the Pde Boundary Conditionmentioning

confidence: 99%

mentioning

confidence: 99%

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Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Windcliff¹,

Forsyth²,

Vetzal³

2004

JCF

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“…Our estimate is essentially based on material presented in Sz.-Nagy and Foia~ [15], [16]. This result is related to a well-known theorem due to von Neumann and recent investigations by Lenferink and Spijker [7], Lubich and Nevanlinna [8], Reddy and Trefethen [ 13]. A similar stability estimate will be presented for multistep discretizations of ( 1.2).…”

Section: II the Purpose Of The Papermentioning

confidence: 76%

Stability estimates based on numerical ranges with an application to a spectral method

Dorsselaer

Hundsdorfer

1994

BIT

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Abstract. CW/ P.O. Box 94079, 1090 GB Amsterdam, The Netherlands willem@cwi.nlThis paper is concerned with the stability of numerical processes for solving initial value problems. We present a stability result which is related to a well-known theorem by von Neumann, but the requirements to be satisfied are less severe and easier to verify.As an illustration we consider a simple convection-diffusion equation. For the spatial discretization we use a spectral collocation method (based on so-called Legendre-Gauss-Lobatto points). We show that the fully discretized numerical process is stable, provided that the temporal step size is bounded by a constant depending only on the convection-diffusion equation, the number of collocation points and the time-stepping method under consideration.A.MS subject classification: 65Ml2, 65L20, 65M70.Key words: stability of numerical processes, initial(-boundary) value problems, ordinary and partial differential equations, one-step and multistep methods, spectral collocation methods.

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“…In this fashion rigorous stability estimates can be derived, e.g., with respect to the maximum norm, in the numerical solution of initial-boundary value problems. For examples of such applications, see [4]. Theorem 2.2 in §2 contains the main result of this paper.…”

Section: Introductionmentioning

confidence: 99%

On a generalization of the resolvent condition in the Kreiss matrix theorem

Lenferink¹,

Spijker²

1991

Math. Comp.

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Abstract. This paper deals with a condition on the resolvent of s x 5 matrices A . In one of the equivalent assertions of the Kreiss matrix theorem, the spectral norm of the resolvent of A at £ must satisfy an inequality for all f lying outside the unit disk in C. We consider a generalization in which domains different from the unit disk and more general norms are allowed.Under this generalized resolvent condition an upper bound is derived for the norms of the «th powers of sxs matrices B . Here, B depends on A via a relation B = is an arbitrary rational function. The upper bound grows linearly with s > 1 and is independent of n > 1 . This generalizes an upper bound occurring in the Kreiss theorem where B = A .Like the classical Kreiss theorem, the upper bound derived in this paper can be used in the stability analysis of numerical methods for solving differential equations.

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On the use of stability regions in the numerical analysis of initial value problems

Cited by 34 publications

References 21 publications

Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Analysis of the stability of the linear boundary condition for the Black–Scholes equation

Stability estimates based on numerical ranges with an application to a spectral method

On a generalization of the resolvent condition in the Kreiss matrix theorem

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