1,868 research outputs found
Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in General Relativity
The description of many dynamical problems like the particle motion in higher
dimensional spherically and axially symmetric space-times is reduced to the
inversion of a holomorphic hyperelliptic integral. The result of the inversion
is defined only locally, and is done using the algebro-geometric techniques of
the standard Jacobi inversion problem and the foregoing restriction to the
--divisor. For a representation of the hyperelliptic functions the
Klein--Weierstra{\ss} multivariable sigma function is introduced. It is shown
that all parameters needed for the calculations like period matrices and
Abelian images of branch points can be expressed in terms of the periods of
holomorphic differentials and theta-constants. The cases of genus two and three
are considered in detail. The method is exemplified by particle motion
associated with a genus three hyperelliptic curve
A New Discontinuous Galerkin Finite Element Method for Directly Solving the Hamilton-Jacobi Equations
In this paper, we improve upon the discontinuous Galerkin (DG) method for
Hamilton-Jacobi (HJ) equation with convex Hamiltonians in (Y. Cheng and C.-W.
Shu, J. Comput. Phys. 223:398-415,2007) and develop a new DG method for
directly solving the general HJ equations. The new method avoids the
reconstruction of the solution across elements by utilizing the Roe speed at
the cell interface. Besides, we propose an entropy fix by adding penalty terms
proportional to the jump of the normal derivative of the numerical solution.
The particular form of the entropy fix was inspired by the Harten and Hyman's
entropy fix (A. Harten and J. M. Hyman. J. Comput. Phys. 50(2):235-269, 1983)
for Roe scheme for the conservation laws. The resulting scheme is compact,
simple to implement even on unstructured meshes, and is demonstrated to work
for nonconvex Hamiltonians. Benchmark numerical experiments in one dimension
and two dimensions are provided to validate the performance of the method
- …