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

Dorsselaer, van J.L.M.; Hundsdorfer, Willem

doi:10.1007/bf01955870

Cited by 5 publications

(2 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let R be any given k × k matrix-valued rational function of one complex variable. The following lemma is a direct consequence of a result by van Dorsselaer and Hundsdorfer [4]. 3k for all complex square matrices A satisfying D(ρ).…”

Section: Lemma 210 Let ρ ∈ [0 ∞) Assume That D[ρ] × D[∞] ⊂ S and That The Imex Linear Multistep Methods Is Stable At Infinity Then There mentioning

confidence: 79%

See 1 more Smart Citation

On the contractivity of implicit–explicit linear multistep methods

Hout

2002

Applied Numerical Mathematics

View full text Add to dashboard Cite

This paper is concerned with the class of implicit-explicit linear multistep methods for the numerical solution of initial value problems for ordinary differential equations which are composed of stiff and nonstiff parts. We study the contractivity of such methods, with regard to linear autonomous systems of ordinary differential equations and a (scaled) Euclidean norm. In addition, we derive a strong stability result based on the stability regions of these methods.

show abstract

Section: Lemma 210 Let ρ ∈ [0 ∞) Assume That D[ρ] × D[∞] ⊂ S and That The Imex Linear Multistep Methods Is Stable At Infinity Then There mentioning

confidence: 79%

“…We remark that the above theorem is related to [4,Theorem 2.6], which deals with the case of linear multistep methods.…”

Section: Strong Stabilitymentioning

confidence: 99%

On the contractivity of implicit–explicit linear multistep methods

Hout

2002

Applied Numerical Mathematics

View full text Add to dashboard Cite

show abstract

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

Straetemans¹

1998

Applied Numerical Mathematics

View full text Add to dashboard Cite

Dispersive and Dissipative Behavior of the Spectral Element Method

Ainsworth¹,

Wajid²

2009

SIAM J. Numer. Anal.

View full text Add to dashboard Cite

If the nodes for the spectral element method are chosen to be the Gauss-Legendre-Lobatto points, then the resulting mass matrix is diagonal and the method is sometimes then described as the Gauss-point mass lumped finite element scheme. We study the dispersive behaviour of the scheme issue in detail and provide both a qualitative description of the nature of the dispersive and dissipative behaviour of the scheme along with precise quantitative statements of the accuracy in terms of the mesh size and the order of the scheme. We prove that (a) the Gausspoint mass lumped scheme (i.e. spectral element method) tends to exhibit phase lag whereas the (consistent) finite element scheme tends to exhibit phase lead; (b) the absolute accuracy of the spectral element scheme is 1/p times better than that of the finite element scheme despite the use of numerical integration; (c) when the order p, the mesh-size h and the frequency of the wave ω satisfy 2p + 1 ≈ ωh the true wave is essentially fully resolved. As a consequence, one obtains a proof of the general rule of thumb sometimes quoted in the context of spectral element methods: π modes per wavelength are needed to resolve a wave.

show abstract

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

Cited by 5 publications

References 15 publications

On the contractivity of implicit–explicit linear multistep methods

On the contractivity of implicit–explicit linear multistep methods

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

Dispersive and Dissipative Behavior of the Spectral Element Method

Contact Info

Product

Resources

About