Formal Proof of a Wave Equation Resolution Scheme: the Method Error
Sylvie Boldo (INRIA Saclay - Ile de France, LRI), Fran\c{c}ois, Cl\'ement (INRIA Rocquencourt), Jean-Christophe Filli\^atre (INRIA Saclay -, Ile de France, LRI), Micaela Mayero (LIPN, INRIA Rh\^one-Alpes / LIP, Laboratoire de l'Informatique du Parall\'elisme)
TL;DR
This paper formalizes and machine-checks the convergence proof of a finite difference scheme for the 1D acoustic wave equation using Coq, addressing challenges in defining asymptotic behaviors.
Contribution
It provides the first formal, machine-checked proof of convergence for a finite difference scheme solving the acoustic wave equation.
Findings
Successful formal proof of convergence in Coq
Addresses asymptotic behavior formalization
First machine-checked proof of this kind
Abstract
Popular finite difference numerical schemes for the resolution of the one-dimensional acoustic wave equation are well-known to be convergent. We present a comprehensive formalization of the simplest one and formally prove its convergence in Coq. The main difficulties lie in the proper definition of asymptotic behaviors and the implicit way they are handled in the mathematical pen-and-paper proofs. To our knowledge, this is the first time such kind of mathematical proof is machine-checked.
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Taxonomy
TopicsUnderwater Acoustics Research · Geophysical Methods and Applications · Seismic Imaging and Inversion Techniques