Investigation of three-level finite-difference time-domain methods for multidimensional acoustics and electromagnetics.

INVESTIGATION OF THREE-LEVEL FINITE-DIFFERENCE TIME-DOMAIN METHODS FOR MULTIDIMENSIONAL ACOUSTICS AND ELECTROMAGNETICS by Brian Thao Nguyen Chair: P.L. Roe This dissertation deals with accurate nite-di erence time-domain methods for e ciently solving acoustics and electromagnetic scattering problems. Problems in 1D, 2D and 3D are treated. Primary emphasis is given to comparing variations of the existing standard leapfrog schemes and new upwind leapfrog schemes and to the e ectiveness of their high-order extensions. Numerical errors are described for the schemes at given resolutions. Afterward, cost-e ciency is compared between schemes dealing with the same problem. In 1D, additional comparison with di usive central-di erence and upstream-biased schemes are made to gain a perspective on the advantages and disadvantages of non-di usive schemes. The multi-dimensional upwind leapfrog schemes are derived on the basis of those originally developed for the 1D advection equation, as many upstream-biased schemes are. To extend the 1D schemes for the multi-dimensional acoustics and electromagnetics equations, the equations are rst written in bicharacteristic forms. Relating the bicharacteristics equations to the 1D advection equation shows that multidimensional upwind leapfrog schemes can be derived by including terms that couple the equations in the di erent directions and by using an appropriate layout of variables on the grid. All schemes surpass the widely-used second-order accurate standard leapfrog scheme in most respect, but the fourth-order extension of this scheme is very competitive. A very tight error tolerance is required to make the sixth-order accurate version feasible. Higher-order schemes are always more memory-e cient than lowerorder versions, but their large computational stencils are less desirable than the more compact lower-order stencils. The upwind leapfrog schemes always uses a smaller stencil than the standard leapfrog scheme stencils, for a given order of accuracy. The 2D fourth-order accurate upwind leapfrog scheme is a highly desirable scheme in most respects. In 3D, the schemes are more closely matched, with the high-order standard leapfrog schemes being better in terms of overall e ciency, though the upwind leapfrog schemes are not grossly ine cient in comparison. No stable highorder upwind leapfrog scheme for electromagnetics has been found. Costs range over nearly two orders of magnitude between the least and most e cient schemes, considering only the bare schemes. However, depending on the particulars of the actual problem being solved, such as the presence of inhomogeneity, the results of the e ciency comparison may change.