A Priori Error Analysis of an Adaptive Splitting Scheme for Non-Autonomous Second-Order Systems

We present a fully discrete splitting-Galerkin scheme for second-order, non-autonomous abstract Cauchy problems with time-dependent perturbations. By reformulating the second-order equation as a first-order system in the product space, we apply a Galerkin semi-discretization in space of order O(hk) and a Strang splitting in time of order O(Δt2). An embedded Runge–Kutta controller provides adaptive time-stepping to handle rapid temporal variations in the perturbation operator B(t). Under standard regularity and commutator assumptions on A(t) and B(t), we establish a priori error estimates max0≤tn≤T∥u(tn)−un∥Z=O(hk+Δt2). Numerical experiments for a 1D perturbed wave equation confirm the theoretical convergence rates, illustrate stability thresholds in the unstable regime, and demonstrate up to 40% savings in computational cost via adaptivity.