Second-Order Fully Discrete Central-Upwind Scheme for Two-Dimensional Hyperbolic Systems of Conservation Laws (original) (raw)

Central-upwind schemes on triangular grids for hyperbolic systems of conservation laws

Numerical Methods for Partial Differential Equations, 2005

We present a family of central-upwind schemes on general triangular grids for solving two-dimensional systems of conservation laws. The new schemes enjoy the main advantages of the Godunov-type central schemes-simplicity, universality, and robustness and can be applied to problems with complicated geometries. The "triangular" central-upwind schemes are based on the use of the directional local speeds of propagation and are a generalization of the central-upwind schemes on rectangular grids, recently introduced in Kurganov et al. [SIAM J Sci Comput 23 (2001), 707-740]. We test a second-order version of the proposed scheme on various examples. The main purpose of the numerical experiments is to demonstrate the potential of our method. The more universal "triangular" central-upwind schemes provide the same high accuracy and resolution as the original, "rectangular" ones, and at the same time, they can be used to solve hyperbolic systems of conservation laws on complicated domains, where the implementation of triangular or mixed grids is advantageous.

Adaptive central-upwind schemes for hyperbolic systems of conservation laws

2006

Central-upwind schemes, recently introduced in [7, 8], are simple, universal, and efficient Godunov-type projectionevolution methods for hyperbolic systems of conservation laws. Their efficiency, however, is typically reduced at the projection step, which requires nonlinear limiters to prevent oscillations. Such limiters, especially those that are higher than second order accurate, are computationally expensive. They may also cause excessive smearing of (liner) contact waves. We present a new adaptive version of the central-upwind schemes, in which second-order limiters are applied near shock waves only, while in the rest of the computational domain, an unlimited fifthorder reconstruction is employed. Our adaption strategy is based on the modified version of the weak local residual from [5], which is used to automatically distinguish between shock and contact discontinuities.

Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations

Siam Journal on Scientific Computing, 2001

We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton-Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720-742].

Adaptive Semidiscrete Central-Upwind Schemes for Nonconvex Hyperbolic Conservation Laws

Siam Journal on Scientific Computing, 2007

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 con- verge to the unique entropy solution or the convergence may be so slow that achieving a proper resolution would require the use of

On the reduction of numerical dissipation in central-upwind schemes

Computer Physics Communications, 2007

We study central-upwind schemes for systems of hyperbolic conservation laws, recently introduced in [A. Kurganov, S. Noelle and G. Petrova, SIAM J. Sci. Comput., 23 (2001), pp. 707‐740]. Similarly to the staggered central schemes, these schemes are central Godunov-type projection-evolution methods that enjoy the advantages of high resolution, simplicity, universality, and robustness. At the same time, the central-upwind framework

Path-Conservative Central-Upwind Schemes for Nonconservative Hyperbolic Systems

ESAIM: Mathematical Modelling and Numerical Analysis

We develop path-conservative central-upwind schemes for nonconservative one-dimensional hyperbolic systems of nonlinear partial differential equations. Such systems arise in a variety of applications and the most challenging part of their numerical discretization is a robust treatment of nonconservative product terms. Godunov-type central-upwind schemes were developed as an efficient, highly accurate and robust ``black-box'' solver for hyperbolic systems of conservation and balance laws. They were successfully applied to a large number of hyperbolic systems including several nonconservative ones. To overcome the difficulties related to the presence of nonconservative product terms, several special techniques were proposed. However, none of these techniques was sufficiently robust and thus the applicability of the original central-upwind schemes was rather limited. In this paper, we rewrite the central-upwind schemes in the form of path-conservative schemes. This helps us (i)...

Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton--Jacobi Equations

SIAM Journal on Scientific Computing, 2001

We introduce new Godunov-type semidiscrete central schemes for hyperbolic systems of conservation laws and Hamilton-Jacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation and can be viewed as a generalization of the schemes from [A. A. Kurganov and G. Petrova, A third-order semidiscrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems, Numer. Math., to appear] and [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 720-742].

High-Order Central Schemes for Hyperbolic Systems of Conservation Laws

Siam Journal on Scientific Computing, 1999

A family of shock capturing schemes for the approximate solution of hyperbolic systems of conservation laws is presented. The schemes are based on a modified ENO reconstruction of pointwise values from cell averages and on approximate computation of the flux on cell boundaries. The use of a staggered grid avoids the need of a Riemann solver. The integral of the fluxes is computed by Simpson's rule. The approximation of the flux on the quadrature nodes is obtained by Runge-Kutta schemes with the aid of natural continuous extension (NCE). This choice gives great flexibility at low computational cost. Several tests are performed on the scalar equation and on systems. The numerical results confirm the expected accuracy and the high resolution properties of the schemes.