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Deng, Zihao

Mathematics and Statistics

2016

M.S.

This thesis presents the stability analysis of the numerical method of characteristic (MoC) that is applied to hyperbolic partial differential equations (PDEs). Simple Euler method, Modified Euler method and Leapfrog method are used for numerical integration along the characteristics. The corresponding MoC schemes are referred to as MoC-SE, MoC-ME, and MoC-LF respectively. We discovered and explained two unusual phenomena. First, certain non-periodic boundary condition (b.c.) could eliminate the numerical instability for some schemes such as the MoC-ME, where the instability exists for periodic b.c. However, it is commonly believed that if von Neumann analysis, i.e., assuming periodic b.c., predicts numerical instability, then there must be numerical instability if one uses non-periodic b.c.. Second, a symplectic method (LF), which is known to work well for energy-preserving ordinary differential equations, introduces a strong numerical instability when integrating energy-preserving PDEs by the MoC. In this thesis, we worked out a new method of analyzing the numerical scheme with non-periodic boundary conditions and explained in details why the non-periodic boundary conditions could eliminate the numerical instability. We also illustrated why our result contradicts the common knowledge that von Neumann stability is necessary for numerical stability of the scheme.