Topological Study of Global Quantization in Non-Abelian Gauge Theories

We study some topological aspects of non-abelian gauge theories intimately connected to the Lie algebras of the gauge groups and the homotopy theory in the generalized gauge orbit space. The physics connection to the non-perturbative solution to strong CP problem as originally proposed by the author is also discussed. Some relevant topological formulas are also given and discussed. A result from the physics application is that the usual gauge orbit space on the compactified space can contain at most a $Z_2$ monopole structure in the SP(2N) gauge theories. Some relevance to the open universe is also discussed. We expect that our results may also be useful to the other studies of non-abelian gauge theories in general.Comment: JSUHEP930701, LaTeX 17 page

High-order accurate well-balanced energy stable adaptive moving mesh finite difference schemes for the shallow water equations with non-flat bottom topography

This paper proposes high-order accurate well-balanced (WB) energy stable (ES) adaptive moving mesh finite difference schemes for the shallow water equations (SWEs) with non-flat bottom topography. To enable the construction of the ES schemes on moving meshes, a reformulation of the SWEs is introduced, with the bottom topography as an additional conservative variable that evolves in time. The corresponding energy inequality is derived based on a modified energy function, then the reformulated SWEs and energy inequality are transformed into curvilinear coordinates. A two-point energy conservative (EC) flux is constructed, and high-order EC schemes based on such a flux are proved to be WB that they preserve the lake at rest. Then high-order ES schemes are derived by adding suitable dissipation terms to the EC schemes, which are newly designed to maintain the WB and ES properties simultaneously. The adaptive moving mesh strategy is performed by iteratively solving the Euler-Lagrangian equations of a mesh adaptation functional. The fully-discrete schemes are obtained by using the explicit strong-stability preserving third-order Runge-Kutta method. Several numerical tests are conducted to validate the accuracy, WB and ES properties, shock-capturing ability, and high efficiency of the schemes.Comment: 40 pages, 16 figure