Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations

Kraaijevanger, J. F. B. M.; Lenferink, H. W. J.; Spijker, M. N.

doi:10.1016/0377-0427(87)90126-9

Cited by 18 publications

(31 citation statements)

References 23 publications

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“…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 demonstrate that the discrete timestepping operator is power-bounded, via numerical experiments.…”

mentioning

confidence: 96%

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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mentioning

confidence: 96%

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

Windcliff¹,

Forsyth²,

Vetzal³

2004

JCF

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“…This was pointed out by, among others, Griffiths, Christie, and Mitchell [11], who displayed an instructive example where one has essentially \vn\ > a"\vQ\ (n = 1, 2, ... , s), \vn\ < a\vQ\ (n > s) with a > 1 and arbitrary dimension 5 > 1 . See also [15,18,22,27,28] [4,6,15,20,25,27]. However, the conditions imposed in these references on h, A , and S are not completely satisfactory in that they cannot be fulfilled in some cases of practical interest.…”

mentioning

confidence: 99%

“…Using this concept, we review in §2.3 stability results from [4,6,15,20,25,27]. In §2.4 we relate the M -numerical range to so-called circle conditions, which were basic for the stability analysis of [15,20,27].…”

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confidence: 99%

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

Lenferink¹,

Spijker²

1991

Math. Comp.

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Abstract. This paper deals with the stability analysis of one-step methods in the numerical solution of initial (-boundary) value problems for linear, ordinary, and partial differential equations. Restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size. These restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration.The errors in the one-step methods are measured in arbitrary norms (not necessarily generated by an inner product).The theory is illustrated in the numerical solution of the heat equation and some other differential equations, where the error growth is measured in the maximum norm.

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“…'s with non-normal matrices has been a major research topic in the numerical analysis community in recent years 8,9,11,12,13,14,15]; A recent review article by van Dorsselaer et al 10] provides an excellent overview of these and many other references. The application is often to families of non-normal matrices arising from spatial discretisations of p.d.e.'s.…”

Section: Introductionmentioning

confidence: 99%

“…Instead, attention has focussed on weaker de nitions of stability which are more easily achieved and are still useful for practical computations. One is algebraic stability 8,11,12] which allows a linear growth in the transient solution of the form kU (n) k n kU (0) k; (2.12) where is again a uniform constant. Another, due to Kreiss and Wu 9], is generalised stability which is based on exponentially weighted integrals over time for a inhomogeneous di erence equation with homogeneous initial conditions.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of Galerkin/Runge-Kutta Navier-Stokes discretisations on unstructured grids

Giles¹

1995

12th Computational Fluid Dynamics Conference

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This paper presents a timestep stability analysis for a class of discretisations applied to the linearised form of the Navier-Stokes equations on a 3D domain with periodic boundary conditions. Using a suitable de nition of the`perturbation energy' it is shown that the energy is monotonically decreasing for both the original p.d.e. and the semi-discrete system of o.d.e.'s arising from a Galerkin discretisation on a tetrahedral grid. Using recent theoretical results concerning algebraic and generalised stability, su cient stability limits are obtained for both global and local timesteps for fully discrete algorithms using Runge-Kutta time integration. Subject classi cations: AMS(MOS): 65M10, 65M20, 65M60, 76-08, 76N10

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Stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations

Cited by 18 publications

References 23 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

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

Stability analysis of Galerkin/Runge-Kutta Navier-Stokes discretisations on unstructured grids

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