Finite difference method

Abstract: We have developed a finite-difference solution for three-dimensional (3-D) transient electromagnetic problems. The solution steps Maxwell's equations in time using a staggered-grid technique. The time-stepping uses a modified version of the Du Fort-Frankel method which is explicit and always stable. Both conductivity and magnetic permeability can be functions of space, and the model geometry can be arbitrarily complicated. The solution provides both electric and magnetic field responses throughout the earth. Because it solves the coupled, first-order Maxwell's equations, the solution avoids approximating spatial derivatives of physical properties, and thus overcomes many related numerical difficulties. Moreover, since the divergence-free condition for the magnetic field is incorporated explicitly, the solution provides accurate results for the magnetic field at late times.An inhomogeneous Dirichlet boundary condition is imposed at the surface of the earth, while a homogeneous Dirichlet condition is employed along the subsurface boundaries. Numerical dispersion is alleviated by using an adaptive algorithm that uses a fourth-order difference method at early times and a second-order method at other times. Numerical checks against analytical, integral-equation, and spectral differential-difference solutions show that the solution provides accurate results.Execution time for a typical model is about 3.5 hours on an IBM 3090/600S computer for computing the field to 10 ms. That model contains 100 X 100 X 50 grid points representing about three million unknowns and possesses one vertical plane of symmetry, with the smallest grid spacing at 10 m and the highest resistivity at 100 Omega . m. The execution time indicates that the solution is computer intensive, but it is valuable in providing much-needed insight about TEM responses in complicated 3-D situations.