Verification of frequency-dependent impedance boundaries for the Finite-Volume Time-Domain scheme

Numerical simulation methods are widely used to approximate solutions to the wave-equation, taking into account the initial condition ('initial sources') and boundary conditions ('wall materials'). Trustworthy simulations are essential; a primary step is to ensure the correctness of the implementation through a process known as code verification. In this work, we do code verification by studying the convergence of the known finite-volume time-domain (FVTD) method with a frequency-dependent boundary condition modeled as an inductance/resistance/capacitance (LRC) termination. Due to the difficulties of obtaining an analytical solution for frequency-dependent impedance boundaries, we thoroughly derive and implement The Method of Manufactured Solutions tailored to the wave-equation inside a rectangular room for uniform frequency- dependent wall impedance. Asymptotic estimates are found empirically based on a convergence study compared against the manufactured ground truth solution. We investigate the impact of the frequency-dependent boundaries on the convergence affected by the total sum of numerical errors and validate the theoretical properties of the FVTD scheme for various LRC model configurations. The method should also serve as a useful resource for others to verify the correctness of their code implementations.