A Stability-Preserving Staggered Finite Difference Approximation for a Hybrid Wave System with Conditional Exponential Stability

Source abstract

We analyze a staggered finite difference approximation for a hybrid PDE-ODE transmission system consisting of two one-dimensional wave equations coupled through a dynamic interior mass. The continuous model is considered in an interface-damped regime in which the right boundary is conservative and the only dissipation acts through the transmission condition. In this regime, exponential stability is conditional and is governed by an arithmetic nonresonance condition on the geometry and physical coefficients. The main question addressed in this paper is whether this conditional stability mechanism is preserved at the semi-discrete level. We construct a finite difference scheme adapted to the dynamic transmission law and prove that it preserves the discrete energy dissipation identity. We then derive a mesh-dependent spectral nonresonance condition characterizing the absence of purely imaginary eigenvalues of the semi-discrete generator. For each fixed mesh, this condition yields exponential stability of the semi-discrete semigroup. Our main result concerns the decay rate under mesh refinement: it shows that, when the travel-time ratio of the two strings is a rational number satisfying the arithmetic condition and the meshes are matched so that the numerical travel time per cell is the same on both strings, the semi-discrete generators satisfy a mesh-uniform resolvent estimate on the imaginary axis; consequently the semi-discrete energy decays exponentially at a rate independent of the mesh size. The proof combines an exact discrete Rellich identity, explicit transfer identities for the two strings, a factorization of the interface force produced by the half-cell inertia, and a parity argument on the matched family. We further prove that the discrete nonresonance condition is branchwise consistent with the continuous arithmetic condition and that resonant discrete eigenvalues converge to the corresponding continuous eigenvalues. Numerical experiments illustrate the resonant and nonresonant regimes, the mesh-independent separation of the computed spectral abscissa from the imaginary axis on matched meshes predicted by the theory, and its loss on unmatched meshes.