  Simulation of 3D low-frequency electromagnetic fields propagating in the
Earth is computationally expensive. We present a fictitious wave domain
high-order finite-difference time-domain (FDTD) modelling method on nonuniform
grids to compute frequency-domain 3D controlled-source electromagnetic (CSEM)
data. The method overcomes the inconsistency issue widely present in the
conventional 2nd order staggered grid finite difference scheme over nonuniform
grid, achieving high accuracy with arbitrarily high order scheme. The
finite-difference coefficients adaptive to the node spacings, can be accurately
computed by inverting a Vandermonde matrix system using efficient algorithm. A
generic stability condition applicable to nonuniform grids is established,
revealing the dependence of the time step and these finite-difference
coefficients. A recursion scheme using fixed point iterations is designed to
determine the stretching factor to generate the optimal nonuniform grid. The
grid stretching in our method reduces the number of grid points required in the
discretization, making it more efficient than the standard high-order FDTD with
a densely sampled uniform grid. Instead of stretching in both vertical and
horizontal directions, better accuracy of our method is observed when the grid
is stretched along the depth without horizontal stretching. The efficiency and
accuracy of our method are demonstrated by numerical examples.
