We give an algorithm to compute $N$ steps of a convolution quadrature approximation to a continuous temporal convolution using only $O(N \log N)$ multiplications and $O(\log N)$ active memory. The method does not require evaluations of the convolution kernel, but instead $O(\log N)$ evaluations of its Laplace transform, which is assumed sectorial.
The algorithm can be used for the stable numerical solution with quasi-optimal complexity of linear and nonlinear integral and integro-differential equations of convolution type. In a numerical example we apply it to solve a subdiffusion equation with transparent boundary conditions.
An error analysis of Runge-Kutta convolution quadrature based on Gauss methods applied to hyperbolic operators is given. The order of convergence relies heavily on the parity of the number of stages, a more favourable situation arising for the odd cases than the even ones. Moreover, for particular situations the order of convergence is higher than for Radau IIA or Lobatto IIIC methods when using the same number of stages. We further investigate an application to transient acoustic scattering where, for certain scattering obstacles, the favourable situation occurs in the important case of the e…
We describe a numerical method for the solution of acoustic exterior scattering problems based on the time-domain boundary integral representation of the solution. As the spatial discretization of the resulting time-domain boundary integral equation we use either the method of fundamental solutions (MFS) or the Galerkin boundary element method (BEM). In time we apply either a standard convolution quadrature (CQ) based on an A-stable linear multistep method or a modified CQ scheme. It is well-known that the standard low-order CQ schemes for hyperbolic problems suffer from strong dissipation and…
High-Order Nyström/Convolution-Quadrature Solution of Time-Domain Scattering from Closed and Open Lipschitz Boundaries with Dirichlet and Neumann Boundary Conditions
The numerical solution of dynamical systems with memory requires the efficient evaluation of Volterra integral operators in an evolutionary manner. After appropriate discretisation, the basic problem can be represented as a matrix-vector product with a lower diagonal but densely populated matrix. For typical applications, like fractional diffusion or large scale dynamical systems with delay, the memory cost for storing the matrix approximations and complete history of the data then would become prohibitive for an accurate numerical approximation. For Volterra-integral operators of convolution …
Generalized impedance boundary conditions are effective, approximate boundary conditions that describe scattering of waves in situations where the wave interaction with the material involves multiple scales. In particular, this includes materials with a thin coating (with the thickness of the coating as the small scale) and strongly absorbing materials. For the acoustic scattering from generalized impedance boundary conditions, the approach taken here first determines the Dirichlet and Neumann boundary data from a system of time-dependent boundary integral equations with the usual boundary int…
Runge--Kutta convolution coercivity and its use for time-dependent boundary integral equations
The Maxwell Institute for Mathematics in the Sciences（麦克斯韦数理科学研究所）；School of Mathematical & Computer Sciences, Heriot-Watt University（赫瑞-瓦特大学数学与计算机科学学院）；Mathematisches Institut, Universität Tübingen（蒂宾根大学数学研究所）
Convolution quadrature methods for time-domain scattering from unbounded penetrable interfaces
Institute for Mathematical and Computational Engineering, School of Engineering and Faculty of Mathematics, Pontificia Universidad Católica de Chile（智利天主教大学工程学院与数学学院计算与数学工程研究所）；Department of Mathematics, Massachusetts Institute of Technology（麻省理工学院数学系）
We present a novel computational scheme to solve acoustic wave transmission problems over composite scatterers, i.e. penetrable obstacles possessing junctions or triple points. Our continuous problem is cast as a multiple traces time-domain boundary integral formulation valid in two and three dimensions. Numerically, our two-dimensional non-conforming spatial discretization uses spectral elements based on second kind Chebyshev polynomials while a convolution quadrature scheme is performed in the complex frequency domain. Computational experiments reveal multistep and multistage convolution qua…
We propose an accurate algorithm for a novel sum-of-exponentials (SOE) approximation of kernel functions, and develop a fast algorithm for convolution quadrature based on the SOE, which allows an order $N$ calculation for $N$ time steps of approximating a continuous temporal convolution integral. The SOE method is constructed by a combination of the de la Vallée-Poussin sums for a semi-analytical exponential expansion of a general kernel, and a model reduction technique for the minimization of the number of exponentials under given error tolerance. We employ the SOE expansion for the finite pa…
Fast convolution with free-space Green's functions
Instituto de Telecomunicaciones y Aplicaciones Multimedia (ITEAM), Universidad Politècnica de València（瓦伦西亚理工大学多媒体应用与电信研究所）；Courant Institute, New York University（纽约大学柯朗数学科学研究所）
In this article, we present an $O(N \log N)$ rapidly convergent algorithm for the numerical approximation of the convolution integral with radially symmetric weakly singular kernels and compactly supported densities. To achieve the reduced computational complexity, we utilize the Fast Fourier Transform (FFT) on a uniform grid of size $N$ for approximating the convolution. To facilitate this and maintain the accuracy, we primarily rely on a periodic Fourier extension of the density with a suitably large period depending on the support of the density. The rate of convergence of the method increa…
