674 research outputs found
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
Macroscopic modeling and simulations of room evacuation
We analyze numerically two macroscopic models of crowd dynamics: the
classical Hughes model and the second order model being an extension to
pedestrian motion of the Payne-Whitham vehicular traffic model. The desired
direction of motion is determined by solving an eikonal equation with density
dependent running cost, which results in minimization of the travel time and
avoidance of congested areas. We apply a mixed finite volume-finite element
method to solve the problems and present error analysis for the eikonal solver,
gradient computation and the second order model yielding a first order
convergence. We show that Hughes' model is incapable of reproducing complex
crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally,
using the second order model, we study numerically the evacuation of
pedestrians from a room through a narrow exit.Comment: 22 page
Review of Summation-by-parts schemes for initial-boundary-value problems
High-order finite difference methods are efficient, easy to program, scales
well in multiple dimensions and can be modified locally for various reasons
(such as shock treatment for example). The main drawback have been the
complicated and sometimes even mysterious stability treatment at boundaries and
interfaces required for a stable scheme. The research on summation-by-parts
operators and weak boundary conditions during the last 20 years have removed
this drawback and now reached a mature state. It is now possible to construct
stable and high order accurate multi-block finite difference schemes in a
systematic building-block-like manner. In this paper we will review this
development, point out the main contributions and speculate about the next
lines of research in this area
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