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An Error Analysis Of Runge-kutta Convolution Quadrature

Lubich. For this purpose, somenew theoretical tools such as tensorial divided differences, summation byparts with Runge-Kutta differences and a calculus for Runge-Kutta dis-cretizations of generalized convolution operators such as an associativityproperty will In Section 4.1 we introduced theKronecker products of matrices and their application to tensors of vectors. However we observe a better convergence rate for ρ= 0 as predictedby our theory, actually the rate coincides with the optimal convergence rate forsmooth and compatible problems with uniform time steps this contact form

Theory 1, 1 (2001), 259-267.Comlab, Oxford, Preprint ContactSitemapImprintPrivacy PolicyHomepage MPI MIS font size: 30.06.2016, 16:11 Cookies help us deliver our services. The choice of νand the smoothness assumption on gimply that ϕin (77)is well defined (cf. (7)). Comput. 32(5), 2964–2994 (2010)MathSciNetMATHCrossRef3.Banjai, L., Lubich, Ch.: An error analysis of Runge-Kutta convolution quadrature. SIAM J.

For a function gwhich is defined in the time interval [0, T ], we intro-duceg(n):= (g(tn,i))si=1 ∈Cs.The time step nis denoted as a superscript for vectors and matrices in ordernot to Numerical examples will illustrate the stable and efficient behavior of the resulting discretization.Do you want to read the rest of this article?Request full-text CitationsCitations0ReferencesReferences16An error analysis of Runge–Kutta convolution quadrature[Show abstract] Math. 9(3–5), 187–199 (1992).

Math. 112(4), 637–678 (2009) MathSciNetMATHCrossRef12. Then, it holds(I−zA)−1≤β0:= 2α0r0V−1∥V∥ ∀ z∈Cwith Re z≤r02.(40)Proof. (i) By using Re 1ζ= (Re ζ)/|ζ|2, we conclude that Ris analytic forall z∈Cwith Re z < r0. Thisallows to employ a summation-by-parts formula which allows to gain negativepowers of z(and hence a faster decay of the integrand for large z) on the expenseof increased smoothness requirements on the We will add more flexibility in the discretization by replacingthe regularization parameter νby a parameter ρ∈N0.

This lack of regularity suggests to use a time grid which is algebraicallygraded towards the origin. The m-th time derivative of function uis denoted by ∂mtuand its evaluation at some time point tkis∂mtu(k):= dmudtm(tk).4 Further, we introduce 1= (1)si=1 and, for vectors v,w∈Cs, the bilinear (notsesquilinear!) formv·w:=sj=1vjwj.We SIAM J. http://link.springer.com/article/10.1007/s00211-011-0378-z Convolution Quadrature and Discretized Operational CalculusI.

Retarded Boundary Integral Equations on theSphere: Exact and Numerical Solution. Numer. Then there exists CR>0 such that |R(z)| ≤ CRfor all z∈Cwith Re z≤34r0. Then,lhs = α(m+1,k)nℓ=k+1v(ℓ)⊗ •·n+1ℓ=k+1C(ℓ)B(k+1)w(k+1) ⊗n+1ℓ=k+2w(j)=α(m+1,k)v(k+1) ·C(k+1)B(k+1) w(k+1)nℓ=k+2v(ℓ)⊗ •·n+1ℓ=k+2C(ℓ)n+1ℓ=k+2w(j)=kj=m+1v(j)·kj=m+1B(j)kj=m+1w(j)××v(k+1) ·C(k+1)B(k+1) w(k+1)××nℓ=k+2v(ℓ)⊗ •·n+1ℓ=k+2C(ℓ)n+1ℓ=k+2w(j)and this is the assertion.Theorem 28 (Associativity) Let a Runge-Kutta method be given which sat-isfies Assumption 4.

Springer-Verlag,Berlin, 2010. For a vector-valued function v∈Vs, we set∥v∥V:= max1≤i≤s∥vi∥Vif no confusion is possible.For a function w∈Cr([0, T ], V )and any interval τ⊂[0, T ], we set|w|Cr(τ,V ):= 1r!supt∈τ∥∂rw(t)∥Vand ∥w∥Cr(τ,V ):= max0≤ℓ≤r|v|Cℓ(τ,V Sauter and A. Let Band Ddenote some normed vector spaces andlet L(B, D) be the space of continuous, linear mappings.

and Lubich Ch.: An error analysis of Runge-Kutta convolution quadrature,BIT, 51(3), 483--496,2011. We employ Theorem 28 with V:= K−ρand W:= K−1ρto obtainK−ρ∂ΘtK−1ρ∂Θtw(n)=nm=0ωn,m (0) e(n−m)⊗s⊗ •·n×ℓ=mA−1∆ℓ(Id) w(m)⊗A−11(n−m)⊗with the identity mapping Id. Not logged in Not affiliated Skip to main content Skip to sections This service is more advanced with JavaScript available, learn more at http://activatejavascript.org Search Home Contact Us Log in Eigenfrequencies of fractal drums.

Lubich, and J. Comput. Differing provisions from the publisher's actual policy or licence agreement may be applicable.This publication is from a journal that may support self archiving.Learn more © 2008-2016 researchgate.net. http://lanprolab.net/an-error/analysis-services-an-error-was-encountered-in-the-transport-layer.php II, vol-ume 14 of Springer Series in Computational Mathematics.

The class of problems under consideration is defined asfollows. Anal., 45(1):37--53,2007. Let w:R≥0→Cbe a function which can be continuouslyextended to R<0by zero.

Numerische Math-ematik, 67(3):365–389, 1994.35 [15] C.

Since Ris a Pad´e approximation of the exponential function Theorem4.12 in [8] implies that the coefficient matrix Ais diagonalizable and alleigenvalues di,1≤i≤s, have strictly positive real part. We setq(k+1) := e(k−m)⊗s⊗ • k×ℓ=mA−1∆ℓWw(m)×A−11(k−m)×.The left-hand side in (96) can be written in the formV∂ΘtW∂Θtw(n)=nk=0km=0ωn,k (0) ωk,m (0) e(n−k)⊗s⊗ •·n×ℓ=kA−1∆ℓVq(k+1) ⊗A−11(n−k)⊗Lem. 27=nm=0ωn,m (0)nk=me(n−m)⊗s⊗ •·· n×ℓ=kA−1∆ℓV◦ k×ℓ=mA−1∆ℓWw(m)⊗A−11(n−m)⊗.Next we apply the tensorial Taylor’s theorem gives us theestimate|R(x+ i y)| ≤ |R(i y)|+4CRr0x∀0≤x≤r0/2 and y∈R.Since A-stability implies |R(i y)| ≤ 1 we conclude that|R(z)| ≤ 1 + CRe z∀z∈Cwith 0 ≤Re z≤r0/2holds. Asin the proof of Lemma 14, we assume in more generality thatϕ(r)(0) = 0 ∀r= 0, . . . , ρ +m−1for some m≤q+ 1.To estimate the near field component we

This approach has been refined and made more accessible by tailoring the necessary mathematical tools, avoiding an excess of generality. SIAM Journal on Numerical Analysis, 47:227–249, 2008.[4] C. The book offers two different approaches for the analysis of these integral equations,...https://books.google.com/books/about/Retarded_Potentials_and_Time_Domain_Boun.html?id=pNf7CwAAQBAJ&utm_source=gb-gplus-shareRetarded Potentials and Time Domain Boundary Integral EquationsMy libraryHelpAdvanced Book SearchEBOOK FROM $27.40Get this book in printSpringer ShopAmazon.comBarnes&Noble.comBooks-A-MillionIndieBoundFind in his comment is here on Num.

Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. Numerical examples from acoustic scattering show that the theory describes accurately the convergence behaviour of Runge–Kutta convolution quadrature for this class of applications. Anal. 28(1), 162–185 (2008)MathSciNetMATHCrossRef9.Laliena A.R., Sayas F.-J.: Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves. Not logged in Not affiliated Skip to main content Skip to sections This service is more advanced with JavaScript available, learn more at http://activatejavascript.org Search Home Contact Us Log in