Formal Proof of a Wave Equation Resolution Scheme: the Method Error

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.

💡 Research Summary

The paper presents a fully mechanized proof of convergence for the simplest finite‑difference scheme applied to the one‑dimensional acoustic wave equation, using the Coq proof assistant. While the convergence of such schemes is well‑known in the numerical analysis literature, the authors argue that the traditional pen‑and‑paper proofs hide several subtleties that become obstacles when one attempts to formalize the argument in a theorem prover. The contribution of the work is threefold.

First, the authors give a precise Coq formalization of the continuous problem and its discretization. The wave equation ∂²u/∂t² = c²∂²u/∂x² is posed on a bounded spatial interval with appropriate initial and boundary conditions. The domain is uniformly discretized with time step Δt and space step Δx, leading to the classic central‑difference update
uⁿ⁺¹ᵢ = 2uⁿᵢ – uⁿ⁻¹ᵢ + λ² (uⁿᵢ₊₁ – 2uⁿᵢ + uⁿᵢ₋₁),
where λ = cΔt/Δx. The Courant‑Friedrichs‑Lewy (CFL) condition λ ≤ 1 is explicitly encoded as a hypothesis.

Second, the paper tackles the formal definition of asymptotic notation and the handling of remainder terms that are usually left implicit in textbook proofs. In Coq, the authors introduce a rigorous definition of “big‑O” and of asymptotic equivalence using the standard ε‑δ formulation for real functions. They also formalize the Taylor expansion with a Lagrange remainder, proving that the truncation error of the scheme is bounded by a constant times (Δt² + Δx²). This requires a careful treatment of higher‑order derivatives, which are introduced as separate Coq constants with appropriate continuity and boundedness assumptions.

Third, the convergence proof itself is carried out by first establishing consistency (the truncation error bound) and then proving stability via an energy method. The discrete energy
Eⁿ = Σᵢ