An error analysis of Runge–Kutta convolution quadrature

An error analysis is given for convolution quadratures based on strongly A-stable Runge–Kutta methods, for the non-sectorial case of a convolution kernel with a Laplace transform that is polynomially bounded in a half-plane. The order of approximation depends on the classical order and stage order of the Runge–Kutta method and on the growth exponent of the Laplace transform. Numerical experiments with convolution quadratures based on the Radau IIA methods are given on an example of a time-domain boundary integral operator.

[1]  Mari Paz Calvo,et al.  Runge–Kutta convolution quadrature methods for well-posed equations with memory , 2007, Numerische Mathematik.

[2]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[3]  E. Hairer,et al.  Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems , 1993 .

[4]  P. P. B. Eggermont,et al.  On the quadrature error in operational quadrature methods for convolutions , 1992 .

[5]  Wolfgang Hackbusch,et al.  Sparse convolution quadrature for time domain boundary integral formulations of the wave equation , 2008 .

[6]  Lehel Banjai,et al.  Rapid Solution of the Wave Equation in Unbounded Domains , 2008, SIAM J. Numer. Anal..

[7]  E. Hairer,et al.  Solving Ordinary Differential Equations I , 1987 .

[8]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[9]  E. Hairer,et al.  Solving Ordinary Differential Equations II , 2010 .

[10]  C. Lubich.  Convolution quadrature and discretized operational calculus. I , 1988 .

[11]  Alexander Ostermann,et al.  RUNGE-KUTTA METHODS FOR PARABOLIC EQUATIONS AND CONVOLUTION QUADRATURE , 1993 .

[12]  E. Hairer,et al.  Stiff and differential-algebraic problems , 1991 .

[13]  C. Lubich.  Convolution Quadrature Revisited , 2004 .

[14]  C. Lubich.  Convolution quadrature and discretized operational calculus. II , 1988 .

[15]  C. Lubich,et al.  On convolution quadrature and Hille-Phillips operational calculus , 1992 .

[16]  P. Monk,et al.  A Finite Difference Delay Modeling Approach to the Discretization of the Time Domain Integral Equations of Electromagnetics , 2008, IEEE Transactions on Antennas and Propagation.

[17]  C. Lubich,et al.  On the multistep time discretization of linear\newline initial-boundary value problems and their boundary integral equations , 1994 .

[18]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[19]  J. Butcher.  The Numerical Analysis of Ordinary Di erential Equa-tions , 1986 .

[20]  Christian Lubich,et al.  Fast and Oblivious Convolution Quadrature , 2006, SIAM J. Sci. Comput..

[21]  Lehel Banjai,et al.  Multistep and Multistage Convolution Quadrature for the Wave Equation: Algorithms and Experiments , 2010, SIAM J. Sci. Comput..

[22]  A. Bamberger et T. Ha Duong,et al.  Formulation variationnelle espace‐temps pour le calcul par potentiel retardé de la diffraction d'une onde acoustique (I) , 1986 .

[23]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[24]  G. Dahlquist.  A special stability problem for linear multistep methods , 1963 .

[25]  Francisco-Javier Sayas,et al.  Theoretical aspects of the application of convolution quadrature to scattering of acoustic waves , 2009, Numerische Mathematik.