The well-known Caputo fractional derivative and the corresponding Caputo fractional integral occur naturally in many equations that model physical phenomena under inhomogeneous media. The relationship between the two fractional terms can be readily obtained by applying the Laplace transform to a given equation. We seek to numerically approximate Caputo fractional integrals using a Taylor series expansion for convolution integrals. This naturally extends into being able to approximate convolution integrals for a wider class of convolution integral kernels $K(t-s)$. One of the main advantages under this approach is the ability to numerically approximate weakly singular kernels, which fail to converge under traditional quadrature methods. We provide stability and convergence analysis for these composite quadratures, which offer optimal convergence for approximating functions in $C^γ[0,T]$, where $α\leq γ\leq 5$ and $0<α< 1$. For the order $γ= 1,2,3,4,5$ scheme, the resulting approximation is $O(τ^γ)$ accurate, where $τ$ is the size of the partition of the time domain. By instead utilizing a fractional Taylor series expansion, we are able to obtain for $γ\in (0,5)-\{1,2,3,4\}$ order scheme, which yields an approximation of $O(τ^γ)$ with a constant dependent on the kernel function which improves the order of convergence. This allows for a far wider class of functions to be approximated, and by strengthening the regularity assumption, we are able to obtain more accurate results. General convolution integrals exhibit these results up to $γ= 2$ without the assumption of $K$ being decreasing. Finally, some numerical examples are presented, which validate our findings.
We study the systematic numerical approximation of Maxwell's equations in dispersive media. Two discretization strategies are considered, one based on a traditional leapfrog time integration method and the other based on convolution quadrature. The two schemes are proven to be equivalent and to preserve the underlying energy-dissipation structure of the problem. The second approach, however, is independent of the number of internal states and allows to handle rather general dispersive materials. Using ideas of fast-and-oblivious convolution quadrature, the method can be implemented efficiently…
The convolution quadrature theory is a systematic approach to analyse the approximation of the Riemann-Liouville fractional operator $I^α$ at node $x_{n}$. In this paper, we develop the shifted convolution quadrature ($SCQ$) theory which generalizes the theory of convolution quadrature by introducing a shifted parameter $θ$ to cover as many numerical schemes that approximate the operator $I^α$ with an integer convergence rate as possible. The constraint on the parameter $θ$ is discussed in detail and the phenomenon of superconvergence for some schemes is examined from a new perspective. For so…
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
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…
The method of Helsing and co-workers evaluates Laplace and related layer potentials generated by a panel (composite) quadrature on a curve, efficiently and with high-order accuracy for arbitrarily close targets. Since it exploits complex analysis, its use has been restricted to two dimensions (2D). We first explain its loss of accuracy as panels become curved, using a classical complex approximation result of Walsh that can be interpreted as "electrostatic shielding" of a Schwarz singularity. We then introduce a variant that swaps the target singularity for one at its complexified parameter pr…
Guangdong Province Key Lab of Computational Science, School of Mathematics and Computational Science, Sun Yat-sen University（中山大学数学与计算科学学院）；Department of Mathematics, Syracuse University（雪城大学数学系）
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…
