Linear stability analysis in the numerical solution of initial value problems

Dorsselaer, Jos L. M. van; Kraaijevanger, J. F. B. M.; Spijker, M. N.

doi:10.1017/s0962492900002361

Cited by 82 publications

(56 citation statements)

References 51 publications

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“…(see, e.g., Dorsselaer et al (1993)), which is a much stronger estimate than (1.2)-(1.3) with K = 1. It is a long-standing problem in how far, also for all fixed K greater than 1, the estimates (1.2)-(1.3) can be sharpened, say to…”

Section: 2mentioning

confidence: 89%

“…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%

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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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Section: 2mentioning

confidence: 89%

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confidence: 99%

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

2002

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“…For each t and x we approximate the true solution to (3.1) by Ui(t), ... , UN_ 1(t))T this leads to the initial value problem (1. 2) with dimension N -1, where A is given by…”

Section: Generalization To Multistep Methodsmentioning

confidence: 99%

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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“…is optimal [11,14,15] and half an order of convergence could be lost. In special situations this loss might not occur (see [11]).…”

Section: And (Ii) For R = 1 Is a Direct Consequence Of The Definitionmentioning

confidence: 99%

Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems

Calvo

Palencia

2001

Math. Comp.

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Abstract.A new strategy to avoid the order reduction of Runge-Kutta methods when integrating linear, autonomous, nonhomogeneous initial boundary value problems is presented. The solution is decomposed into two parts. One of them can be computed directly in terms of the data and the other satisfies an initial value problem without any order reduction. A numerical illustration is given. This idea applies to practical problems, where spatial discretization is also required, leading to the full order both in space and time.

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Linear stability analysis in the numerical solution of initial value problems

Cited by 82 publications

References 51 publications

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

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

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

Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems

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