An Implicit Nodal–Staggered Finite Difference Scheme for Damped Shear Beam Models Without Rotational Inertia

Abstract

We develop and analyze a fully discrete finite difference approximation for a damped shear beam model with mixed boundary conditions and no rotational inertia. The absence of a second‐order time derivative in the rotational equation gives rise to an asymmetric temporal structure. This structural feature motivates a discretization that is compatible with the spatial coupling of the two variables and preserves the underlying energy structure of the continuous problem. We introduce an implicit scheme on interlaced nodal and staggered grids and derive an exact discrete energy identity. The identity separates the physical damping from a nonnegative numerical dissipation generated by the implicit time stepping and yields unconditional stability in the discrete energy norm. Consistency is established with a truncation error of order , and convergence follows from stability and consistency by the Lax–Richtmyer equivalence theorem. We further derive a matrix representation that permits a recursive implementation of the coupled scheme. A manufactured‐solution test confirms the predicted first‐order convergence along the refinement path . Numerical experiments for the homogeneous problem illustrate the distinct long‐time behavior associated with the exponentially and polynomially stable regimes of the continuous model. Because the implicit discretization introduces additional dissipation, these experiments are interpreted qualitatively rather than as preservation of the continuous decay rates.