This book provides a comprehensive analysis of time domain boundary integral equations and their discretisation by convolution quadrature and the boundary element method.
Properties of convolution quadrature, based on both linear multistep and Runge–Kutta methods, are explained in detail, always with wave propagation problems in mind. Main algorithms for implementing the discrete schemes are described and illustrated by short Matlab codes; translation to other languages can be found on the accompanying GitHub page. The codes are used to present numerous numerical examples to give the reader a feeling for the qualitative behaviour of the discrete schemes in practice. Applications to acoustic and electromagnetic scattering are described with an emphasis on the acoustic case where the fully discrete schemes for sound-soft and sound-hard scattering are developed and analysed in detail. A strength of the book is that more advanced applications such as linear and non-linear impedance boundary conditions and FEM/BEM coupling are also covered. While the focus is on wave scattering, a chapter on parabolic problems is included which also covers the relevant fast and oblivious algorithms. Finally, a brief description of data sparse techniques and modified convolution quadrature methods completes the book.
Suitable for graduate students and above, this book is essentially self-contained, with background in mathematical analysis listed in the appendix along with other useful facts. Although not strictly necessary, some familiarity with boundary integral equations for steady state problems is desirable.
Author(s): Lehel Banjai, Francisco-Javier Sayas
Series: Springer Series in Computational Mathematics, 59
Publisher: Springer
Year: 2022
Language: English
Pages: 282
City: Cham
Preface
Contents
List of Symbols
1 Some Examples of Causal Convolutions
1.1 Abel's Equation and Duhamel's Principle
1.2 A Quick Review of Laplace Transforms and Causal Distributions
1.3 Convolution Form of Systems of Linear ODE
1.4 A One Dimensional `Scattering' Problem
1.5 A Transmission Problem for the Acoustic Wave Equation in 3D
2 Convolution Quadrature for Hyperbolic Symbols
2.1 Introduction to Operational Calculus
2.2 Discrete Convolutions by CQ
2.3 Derivation of CQ Using the z Transform
2.4 Yet Another Derivation of CQ
2.5 Linear Multistep CQ Convergence Analysis
2.6 Positivity Conservation of CQ
2.7 Generalisation of CQ to Non-uniform Time-Steps
2.8 Combination with a Galerkin Discretisation
2.9 Hyperbolic Kernels Under Perturbation
3 Algorithms for CQ: Linear Multistep Methods
3.1 Computation of CQ Weights and Evaluation of Convolution
3.2 Solving a Discrete Convolutional System (All-Steps-at-Once)
3.3 Recursive, Marching-on-in-Time Implementation
3.4 Avoiding Limits to Accuracy Due to Round-Off
3.5 Examples
4 Acoustic Scattering in the Time Domain
4.1 Acoustic Scattering by Bounded Obstacles
4.2 Superposition of Spherical Waves
Two-Dimensional Problems
4.3 Construction via the Laplace Domain
Integral Operators in the Laplace Domain
4.4 The Bamberger Ha-Duong Theory
4.5 Coercivity of the Acoustic Calderón Operator
4.6 Boundary Integral Formulation of Sound-Soft Scattering
4.7 Boundary Integral Formulation of Sound-Hard Scattering
4.8 Kirchhoff's Formula and the Direct Method
4.9 Full Discretisation: Sound Soft Scattering
4.10 Full Discretisation: Sound-Hard Scattering
4.11 Absorbing (Linear and Nonlinear) Boundary Conditions
Boundary Integral Representation of the Scattered Field
Full-Discretisation of the Linear Problem
Semi-discretisation in Time in the Nonlinear Case
4.12 Equivalence of CQ for TDBIE to Linear Multistep Discretisation of the PDE
4.13 Choice of the Linear Multistep Method
5 Runge-Kutta CQ
5.1 Implicit Differentiation with RK Methods
5.2 Operator Valued Functions
5.3 Convergence of RK CQ
5.4 Implementation of the RK-CQ Method and Simple Tests
5.5 Accuracy of RK-CQ and Comparison with Linear Multistep CQ
5.6 Combination with a Galerkin Discretisation in Space
5.7 Sound-Soft Scattering Revisited
5.8 Details of Some Technical Proofs
6 Transient Electromagnetism
6.1 Maxwell Equation and the Electric Field Integral Equation
6.2 Maxwell Boundary Integral Operators in the Laplace Domain
6.3 Time-Domain Estimates for Scattering by a Perfect Conductor
6.4 Further Topics
7 Boundary-Field Formulations
7.1 Acoustic Scattering by Non-homogeneous, Penetrable Obstacle
7.2 Coupled Domain/Boundary Integral Formulation
7.3 Fully Implicit, Fully Discretised System
7.4 Implicit/Explicit Time-Discretisation
8 Parabolic Problems
8.1 Parabolic Symbols and High Order Multistep CQ
8.2 Fast and Oblivious CQ
8.3 Integration Contour and Quadrature for Convolution Weights
The Case of the Fractional Integral and Derivative
The General Case
8.4 Sub-Diffusion and Diffusion-Wave Equations
A Computational Example
8.5 Application to Hyperbolic Problems
9 Data Sparse Methods and Other Topics
9.1 Data-Sparse Representation of Integral Operators with Complex Frequencies
Low-Frequency Regime
High-Frequency Regime
9.2 Frequency Domain Data-Sparsity in the Solution of Time-Domain Boundary Integral Equations
9.3 Concurrent Use of Time-Domain Sparsity and Data Sparsity
9.4 Modified CQ Schemes
Application to Galerkin Discretisation of V(s)
A Background Material
A.1 Hilbert Spaces and Operators
A.2 Causal Distributions and Their Laplace Transforms
A.3 Causal Convolutions in the Laplace Domain
A.4 The z Transform
A.5 The Lebesgue Spaces
A.6 Basic Theory of Sobolev Spaces
A.7 Circulant Matrices
A.8 Gauss Quadrature
A.9 Runge-Kutta Methods
A.10 Scattering by a Circle/Sphere
3D Case
2D Case
A.11 Some Software
References
Index
