Showing papers by "Alexander Kurganov published in 2007"

A second-order well-balanced positivity preserving central-upwind scheme for the saint-venant system ∗

Abstract: A family of Godunov-type central-upwind schemes for the Saint-Venant system of shallow water equations has been first introduced in (A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36, 397-425, 2002). Depending on the reconstruction step, the second-order versions of the schemes there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simultaneously. Here, we introduce an improved second-order central-upwind scheme which, unlike its forerun- ners, is capable to both preserve stationary steady states (lake at rest) and to guarantee the positivity of the computed fluid depth. Another novel property of the proposed scheme is its applicability to models with discontinuous bottom topography. We demonstrate these features of the new scheme, as well as its high resolution and robustness in a number of one- and two-dimensional examples.

Adaptive Semidiscrete Central-Upwind Schemes for Nonconvex Hyperbolic Conservation Laws

Abstract: We discover that the choice of a piecewise polynomial reconstruction is crucial in computing solutions of nonconvex hyperbolic (systems of) conservation laws. Using semidiscrete central-upwind schemes, we illustrate that the obtained numerical approximations may fail to converge to the unique entropy solution or the convergence may be so slow that achieving a proper resolution would require the use of (almost) impractically fine meshes. For example, in the scalar case, all computed solutions seem to converge to solutions that are entropy solutions for some entropy pairs. However, in most applications, one is interested in capturing the unique (Kruzhkov) solution that satisfies the entropy condition for all convex entropies. We present a number of numerical examples that demonstrate the convergence of the solutions, computed with the dissipative second-order minmod reconstruction, to the unique entropy solution. At the same time, more compressive and/or higher-order reconstructions may fail to resolve composite waves, typically present in solutions of nonconvex conservation laws, and thus may fail to recover the Kruzhkov solution. In this paper, we propose a simple and computationally inexpensive adaptive strategy that allows us to simultaneously capture the unique entropy solution and to achieve a high resolution of the computed solution. We use the dissipative minmod reconstruction near the points where convexity changes and utilize a fifth-order weighted essentially nonoscillatory (WENO5) reconstruction in the rest of the computational domain. Our numerical examples (for one- and two-dimensional scalar and systems of conservation laws) demonstrate the robustness, reliability, and nonoscillatory nature of the proposed adaptive method.

A New Sticky Particle Method for Pressureless Gas Dynamics

Abstract: We first present a new sticky particle method for the system of pressureless gas dynamics. The method is based on the idea of sticky particles, which seems to work perfectly well for the models with point mass concentrations and strong singularity formations. In this method, the solution is sought in the form of a linear combination of $\delta$-functions, whose positions and coefficients represent locations, masses, and momenta of the particles, respectively. The locations of the particles are then evolved in time according to a system of ODEs, obtained from a weak formulation of the system of PDEs. The particle velocities are approximated in a special way using global conservative piecewise polynomial reconstruction technique over an auxiliary Cartesian mesh. This velocities correction procedure leads to a desired interaction between the particles and hence to clustering of particles at the singularities followed by the merger of the clustered particles into a new particle located at their center of mass. The proposed sticky particle method is then analytically studied. We show that our particle approximation satisfies the original system of pressureless gas dynamics in a weak sense, but only within a certain residual, which is rigorously estimated. We also explain why the relevant errors should diminish as the total number of particles increases. Finally, we numerically test our new sticky particle method on a variety of one- and two-dimensional problems as well as compare the obtained results with those computed by a high-resolution finite-volume scheme. Our simulations demonstrate the superiority of the results obtained by the sticky particle method that accurately tracks the evolution of developing discontinuities and does not smear the developing $\delta$-shocks.