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Computation of the solution for the 2D acoustic pulse propagation

Pavel Bakhvalov

TL;DR

This work provides an efficient, provably accurate method for evaluating the exact solution of the 2D acoustic Cauchy problem with Gaussian initial data, yielding $p'(t,r)$ and $u_r'(t,r)$ at arbitrary $(t,r)$ with absolute error $\varepsilon$. The authors combine three integral representations and an asymptotic series, plus a Taylor expansion about $r=0$, and switch among them according to a domain partition defined by $H$, $R_1$, and $R_2$ to maintain $O(\ln(1/\varepsilon))$ operations. They implement a careful error-control strategy using uniform-step, Gauss–Legendre, and Gauss–Jacobi quadratures, with explicit bounds grounded in Parseval identities and Hermite expansions. The approach is implemented in the ColESo library and validated against high-precision references, achieving robust accuracy across regimes and noting that the most costly step is the original Form 1 due to Bessel evaluations. The result is a practical exact-solution evaluator for aeroacoustic verification that enables reliable convergence studies for high-order schemes while maintaining tight computational budgets.

Abstract

We consider the 2D acoustic system with the Gaussian pulse as the initial data. This case was proposed at the first Workshop on benchmark problems in computational aeroacoustics, and it is commonly used for the verification of numerical methods. We construct an efficient algorithm to evaluate the exact solution for a given time t and distance r. For a precision eps, it takes c*ln(1/eps) operations (the evaluation of a Bessel function counts as one operation) where c does not depend on t and r. This becomes possible by using three different integral representations and an asymptotic series depending on t and r.

Computation of the solution for the 2D acoustic pulse propagation

TL;DR

This work provides an efficient, provably accurate method for evaluating the exact solution of the 2D acoustic Cauchy problem with Gaussian initial data, yielding and at arbitrary with absolute error . The authors combine three integral representations and an asymptotic series, plus a Taylor expansion about , and switch among them according to a domain partition defined by , , and to maintain operations. They implement a careful error-control strategy using uniform-step, Gauss–Legendre, and Gauss–Jacobi quadratures, with explicit bounds grounded in Parseval identities and Hermite expansions. The approach is implemented in the ColESo library and validated against high-precision references, achieving robust accuracy across regimes and noting that the most costly step is the original Form 1 due to Bessel evaluations. The result is a practical exact-solution evaluator for aeroacoustic verification that enables reliable convergence studies for high-order schemes while maintaining tight computational budgets.

Abstract

We consider the 2D acoustic system with the Gaussian pulse as the initial data. This case was proposed at the first Workshop on benchmark problems in computational aeroacoustics, and it is commonly used for the verification of numerical methods. We construct an efficient algorithm to evaluate the exact solution for a given time t and distance r. For a precision eps, it takes c*ln(1/eps) operations (the evaluation of a Bessel function counts as one operation) where c does not depend on t and r. This becomes possible by using three different integral representations and an asymptotic series depending on t and r.
Paper Structure (13 sections, 9 theorems, 108 equations, 1 figure)

This paper contains 13 sections, 9 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of the algorithm
  3. Series as $r \rightarrow 0$, $t \rightarrow \infty$
  4. Integral expressions
  5. Accuracy of the numerical integration
  6. General estimates
  7. The case $r > t+H$
  8. Approximation of \ref{['eq_form1']}. The case $t+r \leqslant 1.05H$
  9. Appoximation of \ref{['eq_tamnew_2r']}--\ref{['eq_tamnew_3r']}
  10. The case $t>r+1.152H$, $r>R_1$
  11. The case $r-H < t < r+1.152 H$, $t+r > 1.05H$, $r \geqslant R_2$
  12. Approximation of \ref{['eq_puls_bf']}--\ref{['eq_jjn']}
  13. Verification

Key Result

Lemma 1

There holds $\vert\mathcal{I}_n(t)\vert \leqslant \sqrt{\pi/2}\ 2^{n/2}\ \Gamma(1+n/2)$.

Figures (1)

  • Figure 1: Solution form used depending on $t$ and $r$. 0: $p',u_r'\approx 0$; 1: \ref{['eq_form1']}; 2: series; 3, 4: \ref{['eq_tamnew_2r']}--\ref{['eq_tamnew_3r']}; 5: \ref{['eq_puls_bf']}--\ref{['eq_jjn']}

Theorems & Definitions (13)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • Lemma 6
  • proof
  • ...and 3 more