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Traditional techniques for computing electromagnetic solutions in the time domain rely on finite differences. These so-called FDTD (finite-difference time-domain) methods are usually defined only on regular lattices of points and can be too restrictive for geometrically demanding problems. Great geometric flexibility can be achieved by abandoning the regular latticework of sample points and adopting an unstructured grid. An unstructured grid allows one to place the grid points anywhere one chooses, so that curved boundaries can be fit with ease and local regions in which the field gradients are steep can be selectively resolved with a fine mesh. In this paper we present a technique for solving Maxwell's equations on an unstructured grid based on the Taylor-Galerkin finite-element method. We present several numerical examples which reveal the fundamental accuracy and adaptability of the method. Although our examples are in two dimensions, the techniques and results generalize readily to 3D.

The nonlinear progressive wave equation (NPE) is used to investigate propagation of acoustic pulses in a shallow ocean waveguide. The nonlinearity is shown to affect transmission and reflection at a fluid–fluid interface. It is shown that one effect of the nonlinearity is to reduce the critical angle for total internal reflection from that of the linear case. When loss versus range for a simple isovelocity waveguide is compared with linear normal mode and parabolic equation calculations, the nonlinear result shows increased transmission loss as long as the wave amplitude is significant (an expected result of shock processes), and increased bottom penetration. Nonlinear aging of the waveform alters the excitation of linear modes in the farfield.