An ADI-PE Approach for Modeling Radio Transmission Loss in Tunnels

Alternate direction implicit (ADI) method is used to study radio wave propagation in tunnels using the parabolic equation (PE). We formulate the ADI technique for use in tunnels with rectangular, circular and arched cross sections and with lossy walls. The electrical parameters of the lossy walls are characterized by an equivalent surface impedance. A vector PE is also formulated for use in tunnels with lossy walls. It is shown that the ADI is more computationally efficient than the Crank Nicolson method. However, boundary conditions become more difficult to model. The boundary conditions of the ADI intermediate planes are given the same boundary conditions as the physical plane and the overall accuracy is reduced. Also, when implementing the ADI in tunnels with circular cross sections the order at which the line by line decomposition occurs becomes important. To validate the ADI-PE, we show simulation results for tunnel test cases with known analytical solutions. Furthermore, the ADI-PE is used to simulate real tunnels in order to compare with experimental data. It is shown that the PE models the electric fields most accurately in real tunnels at large distances, where the lower order modes dominate.