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National Repository of Grey Literature 2 records found  Search took 0.00 seconds. 
On the Dijkstra's algorithm in the pedestrian flow problem
 Petrášová, Tereza ; Felcman, Jiří (advisor) ; Kučera, Václav (referee)
Title: On the Dijkstra's algorithm in the Pedestrian Flow Problem Author: Tereza Petrášová Department: Department of Numerical Mathematics Supervisor: doc. RNDr. Jiří Felcman, CSc., Department of Numerical Mathe- matics Abstract: The pedestrian flow problem is described by a coupled system of the first order hyperbolic partial differential equations with the source term and by the functional minimization problem for the desired direction of motion. The functional minimization is based on the modified Dijkstra's algorithm used to find the minimal path to the exit. The original modification of the Dijkstra's algorithm is proposed to increase its efficiency in the pedestrian flow problem. This approach is compared with the algorithm of Bornemann and Rasch for determination of the desired direction of motion based on the solution of the so- called Eikonal equation. Both approaches are numerically tested in the framework of two splitting algorithms for solution of the coupled problem. The former splitting algorithm is based on the finite volume method yielding for the given time instant the piecewise constant approximation of the solution. The latter one uses the implicit discretization by a space-time discontinuous Galerkin method based on the discontinuous piecewise polynomial approximation. The numerical examples...
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Numerical Analysis of a polydisperse sedimentation problem
 Dvořák, Daniel ; Felcman, Jiří (advisor) ; Feistauer, Miloslav (referee)
The problem of the polydisperse sedimentation as the system of the partial differential equations is formulated. The hyperbolicity of the problem and the determination of the eigenvalues of the Jacobi matrix of the flux function is studied. Based on the conservation laws of the mass and momentum completed by the constitutive relations the so called MLB model is derived. The one- dimensional problem is formulated. The Sherman-Morrison formula is used to find the inverse matrix of the sum of the diagonal matrix and the matrix being the product of two vectors. In order to find the eigenvalues of the Jacobi matrix of the flux function the rank two perturbation of the diagonal matrix is used. In such a way the problem of the determination of the eigenvalues is reformulated as the solution of the so called secular equation. The eigenvalues can be localized and the strong hyperbolicity of the problem under certain conditions is proved. 1
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