Conventional and asymptotic stabilities of decomposed compact methods for solving highly oscillatory wave problems.

Abstract

This dissertation explores the numerical stabilities of decomposed compact finite difference methods for solving Helmholtz partial differential equation problems with large wave numbers. It is known that many practically effective approximations to highly oscillatory wave solutions are not stable in the traditional von Neumann sense. We will consider three algorithms that numerically approximate Helmholtz problems. In each case, we investigate conventional stability, as well as the physically relevant asymptotic stability where appropriate. For each scheme constructed, radially symmetric transverse fields and standard polar coordinates are considered. A decomposition is implemented to remove the anticipated singularity in the transverse direction for both the paraxial and nonparaxial Helmholtz equations. Compact scheme structures are introduced in the transverse direction to raise the accuracy and efficiency of the schemes developed for laser and subwavelength optical computations. The first scheme approximates a paraxial Helmholtz problem. It is shown that the numerical method introduced is not only highly accurate and efficient due to its straightforward algorithmic structure, but also conventionally stable under reasonable constraints for practical applications. The second and third schemes approximate nonparaxial Helmholtz problems, with the latter being a dual-scale algorithm. While the highly accurate compact algorithms shy away from the conventional stability, they are shown to be asymptotically stable with index one, facilitating the algorithmic effectiveness, reliability, and applicability. Numerical experiments further demonstrate the high reliability of each scheme when implemented in highly oscillatory optical self-focusing optical beam propagation simulations. Computational examples are presented to illustrate our conclusions. Unless otherwise stated, as in Chapter Five, any appropriate matrix norm may be considered.