Analytical Benchmark Verification of Central Difference Time Integration and Explicit FEM Models for the One-Dimensional Wave Equation

This paper presents an analytical and numerical benchmark verification framework for explicit time integration procedures applied to the one-dimensional wave equation. The study combines an exact analytical solution, a closed-form discrete central-difference solution, and explicit finite element models implemented in LS-DYNA and in the in-house academic FEM code PAK-Multiphysics. Two benchmark problems with the same first sinusoidal spatial mode and homogeneous Dirichlet boundary conditions are considered. The first problem, defined by sinusoidal initial displacement and zero initial velocity, is used to analyse the central-difference discretization and its convergence behaviour. The closed-form discrete response enables direct comparison with the analytical solution and confirms the expected second-order accuracy under coupled mesh and time-step refinement at a fixed Courant number. The second problem, defined by zero initial displacement and sinusoidal initial velocity, introduces a phase-shifted temporal response suitable for explicit finite element verification. Four meshes are analysed over five periods of the first longitudinal mode. The numerical responses are assessed using displacement histories, maximum absolute errors, RMS errors, relative RMS errors, amplitude errors, and an energy check. The PAK-Multiphysics results show very close agreement with the analytical solution and a systematic reduction of the error measures, consistent with the lumped-mass central-difference formulation. The LS-DYNA results provide an independent commercial-code comparison, showing decreasing displacement errors under refinement and bounded total-energy variation. The proposed framework provides a transparent and reproducible benchmark for verifying one-dimensional explicit wave propagation models.