189 research outputs found
Invariant Solutions of the Two-Dimensional Shallow Water Equations with a Particular Class of Bottoms
The two-dimensional shallow water equations with a particular bottom and the
Coriolis's force are studied in this paper. The main goal of
the paper is to describe all invariant solutions for which the reduced system
is a system of ordinary differential equations. For solving the systems of
ordinary differential equations we use the sixth-order Runge-Kutta method.Comment: 7 pages, 9 figures, Submitted to AIP Conference Proceedings 2164,
050003 (2019
Invariant Finite-Difference Schemes for Cylindrical One-Dimensional MHD Flows with Conservation Laws Preservation
On the basis of the recent group classification of the one-dimensional
magnetohydrodynamics (MHD) equations in cylindrical geometry, the construction
of symmetry-preserving finite-difference schemes with conservation laws is
carried out. New schemes are constructed starting from the classical completely
conservative Samarsky-Popov schemes. In the case of finite conductivity,
schemes are derived that admit all the symmetries and possess all the
conservation laws of the original differential model, including previously
unknown conservation laws. In the case of a frozen-in magnetic field (when the
conductivity is infinite), various schemes are constructed that possess
conservation laws, including those preserving entropy along trajectories of
motion. The peculiarities of constructing schemes with an extended set of
conservation laws for specific forms of entropy and magnetic fluxes are
discussed.Comment: 29 pages; some minor fixes and generalizations + Appendix containing
an additional numerical schem
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