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 Grenoble 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 machine-checked formal proof of convergence for a finite difference scheme solving the wave equation.
Findings
Successful formal proof of scheme convergence in Coq
Clarification of asymptotic behavior handling in formal proofs
First machine-checked proof of this kind of mathematical verification
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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