Runge–Kutta convolution quadrature methods for well-posed equations with memory

Calvo, M. P.; Cuesta, Eduardo; Palencia, C.

doi:10.1007/s00211-007-0107-9

Cited by 35 publications

(31 citation statements)

References 21 publications

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“…We only have to change the constants in (5), that depend on these coefficients and translate their influence to all other constants in the Lemma.…”

Section: Bounds In the Resolvent Setmentioning

confidence: 99%

“…We are just going to explain here the scalar-valued schemes, associated to multistep methods. For RK-based methods, we refer the reader to the original article [29], to the more recent [5] and, in the context of waves and integral equations, to [23].…”

Section: Appendix 1: Convolution Quadraturementioning

confidence: 99%

“…For the analysis of Runge-Kutta methods for parabolic equations, the CQ technique was extended to a new class of methods based on RK schemes [29]. The analysis of this class of methods has recently been extended in [5]. The relevant paper for our purposes is [27], where the first example of a CQ method for a retarded integral equation is proposed and analyzed.…”

mentioning

confidence: 99%

“…Proposition 17 DtN − , −DtN + , V −1 ∈ E(1, −Arg, H 1/2 ( )).Proof Let φ ∈ H 1/2 ( ) and define u as the solution tou − s 2 u = 0, in − , γ u = φ, on , so that ∂ ν u = DtN − (s)φ.Then A − s)u, u = DtN − (s)φ, φ and hence by(7),(5) and the trace theoremRe e −ıArg s DtN − (s)φ, φ =…”

mentioning

confidence: 99%

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Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

2009

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In this paper we address several theoretical questions related to the numerical approximation of the scattering of acoustic waves in two or three dimensions by penetrable non-homogeneous obstacles using convolution quadrature (CQ) techniques for the time variable and coupled boundary element method/finite element method for the space variable. The applicability of CQ to waves requires polynomial type bounds for operators related to the operator − s 2 in the right half complex plane. We propose a new systematic way of dealing with this problem, both at the continuous and semidiscrete-in-space cases. We apply the technique to three different situations: scattering by a group of sound-soft and -hard obstacles, by homogeneous and non-homogeneous obstacles.

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“…We only have to change the constants in (5), that depend on these coefficients and translate their influence to all other constants in the Lemma.…”

Section: Bounds In the Resolvent Setmentioning

confidence: 99%

Section: Appendix 1: Convolution Quadraturementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

2009

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“…For BDF-2 we have γ (z) = 3/2 − 2z + z 2 /2. For Runge-Kutta method applied instead of linear multistep method we obtain (based on [5][6][7]):…”

Section: -P2mentioning

confidence: 99%

A combined application of boundary-element and Runge-Kutta methods in three-dimensional elasticity and poroelasticity

Игумнов

Ipatov

Белов

et al. 2015

EPJ Web of Conferences

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Abstract. The report presents the development of the time-boundary element methodology and a description of the related software based on a stepped method of numerical inversion of the integral Laplace transform in combination with a family of Runge-Kutta methods for analyzing 3-D mixed initial boundary-value problems of the dynamics of inhomogeneous elastic and poro-elastic bodies. The results of the numerical investigation are presented. The investigation methodology is based on directapproach boundary integral equations of 3-D isotropic linear theories of elasticity and poroelasticity in Laplace transforms. Poroelastic media are described using Biot models with four and five base functions. With the help of the boundary-element method, solutions in time are obtained, using the stepped method of numerically inverting Laplace transform on the nodes of Runge-Kutta methods. The boundary-element method is used in combination with the collocation method, local element-byelement approximation based on the matched interpolation model. The results of analyzing wave problems of the effect of a non-stationary force on elastic and poroelastic finite bodies, a poroelastic half-space (also with a fictitious boundary) and a layered half-space weakened by a cavity, and a half-space with a trench are presented. Excitation of a slow wave in a poroelastic medium is studied, using the stepped BEM-scheme on the nodes of Runge-Kutta methods.

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On temporal regularity and polynomial decay of solutions for a class of nonlinear time‐delayed fractional reaction–diffusion equations

Thu,

Van Tuan

2024

Mathematische Nachrichten

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This paper is devoted to analyzing the regularity in time and polynomial decay of solutions for a class of fractional reaction–diffusion equations (FrRDEs) involving delays and nonlinear perturbations in a bounded domain of . By establishing some regularity estimates in both time and space variables of the resolvent operator, we present results on the Hölder and ‐regularity in time of solutions for both time‐delayed linear and semilinear FrRDEs. Based on the aforementioned results, we study the existence, uniqueness, and regularity of solutions to an identification problem subjected to the delay FrRDE and the additional observations given at final time. Furthermore, under quite reasonable assumptions on nonlinear perturbations and the technique of measure of noncompactness, the existence of decay solutions with polynomial rates for the problem under consideration is shown.

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Runge–Kutta convolution quadrature methods for well-posed equations with memory

Cited by 35 publications

References 21 publications

Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves

A combined application of boundary-element and Runge-Kutta methods in three-dimensional elasticity and poroelasticity

On temporal regularity and polynomial decay of solutions for a class of nonlinear time‐delayed fractional reaction–diffusion equations

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