Showing papers by "Alexander Kurganov published in 2009"

Central-Upwind Schemes for Two-Layer Shallow Water Equations

Abstract: We derive a second-order semidiscrete central-upwind scheme for one- and two-dimensional systems of two-layer shallow water equations. We prove that the presented scheme is well-balanced in the sense that stationary steady-state solutions are exactly preserved by the scheme and positivity preserving; that is, the depth of each fluid layer is guaranteed to be nonnegative. We also propose a new technique for the treatment of the nonconservative products describing the momentum exchange between the layers. The performance of the proposed method is illustrated on a number of numerical examples, in which we successfully capture (quasi) steady-state solutions and propagating interfaces.

Non-oscillatory central schemes for traffic flow models with arrhenius look-ahead dynamics

Abstract: We first develop non-oscillatory central schemes for a traffic flow model with Arrhenius look-ahead dynamics, proposed in [ A. Sopasakis and M.A. Katsoulakis, SIAM J. Appl. Math., 66 (2006), pp. 921--944]. This model takes into account interactions of every vehicle with other vehicles ahead ("look-ahead'' rule) and can be written as a one-dimensional scalar conservation law with a global flux. The proposed schemes are extensions of the non-oscillatory central schemes, which belong to a class of Godunov-type projection-evolution methods. In this framework, a solution, computed at a certain time, is first approximated by a piecewise polynomial function, which is then evolved to the next time level according to the integral form of the conservation law. Most Godunov-type schemes are based on upwinding, which requires solving (generalized) Riemann problems. However, no (approximate) Riemann problem solver is available for conservation laws with global fluxes. Therefore, central schemes, which are Riemann-problem-solver-free, are especially attractive for the studied traffic flow model. Our numerical experiments demonstrate high resolution, stability, and robustness of the proposed methods, which are used to numerically investigate both dispersive and smoothing effects of the global flux. We also modify the model by Sopasakis and Katsoulakis by introducing a more realistic, linear interaction potential that takes into account the fact that a car's speed is affected more by nearby vehicles than distant (but still visible) ones. The central schemes are extended to the modified model. Our numerical studies clearly suggest that in the case of a good visibility, the new model yields solutions that seem to better correspond to reality.

Fast explicit operator splitting method for convection-diffusion equations

Abstract: SUMMARY Systems of convection‐diffusion equations model a variety of physical phenomena which often occur in real life. Computing the solutions of these systems, especially in the convection dominated case, is an important and challenging problem that requires development of fast, reliable and accurate numerical methods. In this paper, we propose a second-order fast explicit operator splitting (FEOS) method based on the Strang splitting. The main idea of the method is to solve the parabolic problem via a discretization of the formula for the exact solution of the heat equation, which is realized using a conservative and accurate quadrature formula. The hyperbolic problem is solved by a second-order finite-volume Godunov-type scheme. We provide a theoretical estimate for the convergence rate in the case of one-dimensional systems of linear convection‐diffusion equations with smooth initial data. Numerical convergence studies are performed for one-dimensional nonlinear problems as well as for linear convection‐diffusion equations with both smooth and nonsmooth initial data. We finally apply the FEOS method to the one- and two-dimensional systems of convection‐diffusion equations which model the polymer flooding process in enhanced oil recovery. Our results show that the FEOS method is capable to achieve a remarkable resolution and accuracy in a very efficient manner, that is, when only few splitting steps are performed. Copyright q 2006 John Wiley & Sons, Ltd.

The Order of Accuracy of Quadrature Formulae for Periodic Functions

Abstract: The trapezoidal quadrature rule on a uniform grid has spectral accuracy when integrating C ∞ periodic function over a period. The same holds for quadrature formulae based on piecewise polynomial interpolations. In this paper, we prove that these quadratures applied to \({\rm{W}}_{{\rm{per}}}^{{\rm{r,p}}} \) periodic functions with r > 2 and p ≥ 1 have error \({\mathcal O}((\Delta x)^r)\). The order is independent of p, sharp, and for p < ∞ is higher than predicted by best trigonometric approximation. For p=1 it is higher by 1.

Computing MultivaluedSolutions of PressurelessGas Dynamics by Deterministic Particle Methods

Abstract: We compute multivalued solutions of one- and two-dimensional pressure- less gas dynamics equations by deterministic particle methods. Point values of the computed solutions are to be recovered from their singular particle approximations using some smoothing procedure. We study several recovery strategies and demon- strate ability of the particle methods to achieve high resolution.

Quasi-Lagrangian Acceleration of Eulerian Methods

Abstract: We present a simple and efficient strategy for the acceleration of explicit Eulerian methods for multidimensional hyperbolic systems of conservation laws. The strategy is based on the Galilean invariance of dynamic equations and optimization of the reference frame, in which the equations are numerically solved. The optimal reference frame moves (locally in time) with the average characteristic speed of the system, and, in this sense, the resulting method is quasi-Lagrangian. This leads to the acceleration of the numerical computations thanks to the optimal CFL condition and automatic adjustment of the computational domain to the evolving part of the solution. We show that our quasi-Lagrangian acceleration procedure may also reduce the numerical dissipation of the underlying Eulerian method. This leads to a significantly enhanced resolution, especially in the supersonic case. We demonstrate a great potential of the proposed method on a number of numerical examples. AMS subject classifications: 76M12, 65B99, 35L65