280 research outputs found
Fast and oblivious convolution quadrature
We give an algorithm to compute steps of a convolution quadrature
approximation to a continuous temporal convolution using only
multiplications and active memory. The method does not require
evaluations of the convolution kernel, but instead 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
Runge-Kutta convolution coercivity and its use for time-dependent boundary integral equations
A coercivity property of temporal convolution operators is an essential tool in the analysis of time-dependent boundary integral equations and their space and time discretisations. It is known that this coercivity property is inherited by convolution quadrature time discretisation based on A-stable multistep methods, which are of order at most two. Here we study the ques- tion as to which Runge–Kutta-based convolution quadrature methods inherit the convolution coercivity property. It is shown that this holds without any restriction for the third-order Radau IIA method, and on permitting a shift in the Laplace domain variable, this holds for all algebraically stable Runge– Kutta methods and hence for methods of arbitrary order. As an illustration, the discrete convolution coercivity is used to analyse the stability and convergence properties of the time discretisation of a non-linear boundary integral equation that originates from a non-linear scattering problem for the linear wave equation. Numerical experiments illustrate the error behaviour of the Runge–Kutta convolution quadrature time discretisation
Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence
Abstract. We unify the formulation and analysis of Galerkin and Runge–Kutta methods for the time discretization of parabolic equations. This, together with the concept of reconstruction of the approximate solutions, allows us to establish a posteriori superconvergence estimates for the error at the nodes for all methods. 1
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