As shown above, driving a single displacement sample in the FDTD corresponds to driving a velocity input at position on two alternating subgrids over time. Therefore, the filter acts as the filter on either subgrid alone--a first-order difference. Since displacement is being simulated, velocity inputs must be numerically integrated. The first-order difference can be seen as canceling this integration, thereby converting a velocity input to a displacement input, as in Eq.(E.23).
FDTD State Space Model