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A $p$-version of convolution quadrature in wave propagation

Alexander Rieder

Abstract

We consider a novel way of discretizing wave scattering problems using the general formalism of convolution quadrature, but instead of reducing the timestep size ($h$-method), we achieve accuracy by increasing the order of the method ($p$-method). We base this method on discontinuous Galerkin timestepping and use the Z-transform. We show that for a certain class of incident waves, the resulting schemes observes(root)-exponential convergence rate with respect to the number of boundary integral operators that need to be applied. Numerical experiments confirm the findings.

A $p$-version of convolution quadrature in wave propagation

Abstract

We consider a novel way of discretizing wave scattering problems using the general formalism of convolution quadrature, but instead of reducing the timestep size (-method), we achieve accuracy by increasing the order of the method (-method). We base this method on discontinuous Galerkin timestepping and use the Z-transform. We show that for a certain class of incident waves, the resulting schemes observes(root)-exponential convergence rate with respect to the number of boundary integral operators that need to be applied. Numerical experiments confirm the findings.
Paper Structure (22 sections, 27 theorems, 130 equations, 4 figures, 1 table)

This paper contains 22 sections, 27 theorems, 130 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Model problem
  3. A brief summary of time domain boundary integral equations
  4. Discontinuous Galerkin timestepping
  5. The $p$-version of Convolution Quadrature
  6. Derivation of the $p$-CQ method
  7. Basic properties of $p-CQ$
  8. Relation to Runge-Kutta CQ
  9. Abstract wave propagation
  10. DG discretizations of wave problems
  11. The CQ scattering problem
  12. The Dirichlet problem
  13. Equivalence principle and the discrete representation formula
  14. Practical aspects
  15. Functional calculus
  16. ...and 7 more sections

Key Result

Proposition 2.2

Assume that we are in one of the following cases: Then the function $u$ defined by the potentials satisfies the model problem eq:wave_eqn_classic.

Figures (4)

  • Figure 7.1: Numerical investigation of $\sigma(\boldsymbol \delta(z)/h)$
  • Figure 8.1: Convergence for the scalar model problem using different windowing functions
  • Figure 8.2: Geometry and solution for the $3d$ example
  • Figure 8.3: Convergence for the $3d$ scattering problem

Theorems & Definitions (64)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Proposition 3.1: SS00b
  • proof
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 4.1
  • ...and 54 more