On the stability and convergence of discretizations of initial value p.d.e.s

Giles, Michael B.

doi:10.1093/imanum/17.4.563

Cited by 7 publications

(6 citation statements)

References 9 publications

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“…Weaker algebraic stability yields convergence only for certain initial data. For a further discussion of these results, see [5].…”

Section: Roughly 10mentioning

confidence: 96%

“…To avoid algebraic complication, we will assume that any Dirichlet boundary conditions are time independent (it is straightforward to extend the analysis to the time-dependent case, at the expense of notational complexity). We can write the discrete equations (3.4) as 5) where the matrix B is defined 1 as…”

Section: The Linear Boundary Conditionmentioning

confidence: 99%

“…This subtle distinction is important since the Lax Equivalence Theorem states that, for linear problems, strong stability is a sufficient and necessary condition for the convergence of a consistent discretization for all initial data. For a discussion on the relationship of algebraic stability to the usual notions of stability and convergence see [5].…”

Section: Non-legitimacy Of the Pde Boundary Conditionmentioning

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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“…Weaker algebraic stability yields convergence only for certain initial data. For a further discussion of these results, see [5].…”

Section: Roughly 10mentioning

confidence: 96%

Section: The Linear Boundary Conditionmentioning

confidence: 99%

Section: Non-legitimacy Of the Pde Boundary Conditionmentioning

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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“…Together with absolute stability which ensures eventual exponential decay of the transient solution, this provides very practical stability criteria. It can also be proved that it provides a sufficient condition for convergence of consistent discretisations of initial value p.d.e.s, subject to very mild restrictions on the smoothness of the initial conditions [8]. This shows the practical usefulness of the concept of algebraic stability.…”

Section: Fully Discrete Equationsmentioning

confidence: 85%

Stability Analysis of Preconditioned Approximations of the Euler Equations on Unstructured Meshes

Moinier

Giles

2002

Journal of Computational Physics

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“…Kreiss (1962) established an important theorem, called the Kreiss matrix theorem, which has been one of the fundamental results for establishing numerical stability. Still recently, much research was devoted to this theorem and variants thereof (see, e.g., Giles (1997), Kraaijevanger (1994), Lubich & Nevanlinna (1991), Reddy & Trefethen (1992), Spijker & Straetemans (1996, Strikwerda & Wade (1991), Toh & Trefethen (1999), and the review papers Borovykh & Spijker (2000), Dorsselaer et al (1993), Nevanlinna (1997), Strikwerda & Wade (1997)). …”

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confidence: 99%

About the sharpness of the stability estimates in the Kreiss matrix theorem

2002

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Abstract. One of the conditions in the Kreiss matrix theorem involves the resolvent of the matrices A under consideration. This so-called resolvent condition is known to imply, for all n ≥ 1, the upper bounds A n ≤ eK(N + 1) and A n ≤ eK(n + 1). Here · is the spectral norm, K is the constant occurring in the resolvent condition, and the order of A is equal to N + 1 ≥ 1.It is a long-standing problem whether these upper bounds can be sharpened, for all fixed K > 1, to bounds in which the right-hand members grow much slower than linearly with N + 1 and with n + 1, respectively. In this paper it is shown that such a sharpening is impossible. The following result is proved: for each > 0, there are fixed values C > 0, K > 1 and a sequence of (N + 1) × (N + 1) matrices A N , satisfying the resolvent condition, such thatThe result proved in this paper is also relevant to matrices A whose -pseudospectra lie at a distance not exceeding K from the unit disk for all > 0.

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On the stability and convergence of discretizations of initial value p.d.e.s

Cited by 7 publications

References 9 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 Analysis of Preconditioned Approximations of the Euler Equations on Unstructured Meshes

About the sharpness of the stability estimates in the Kreiss matrix theorem

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