Central schemes offer a simple and versatile approach for computing approximate solutions of non-linear systems of hyperbolic conservation laws and related PDEs. The solution of such problems often involves the spontaneous evolution of steep gradients. The multiscale aspect of these gradients poses a main computational challenge for their numerical solution. Central schemes utilize a minimal amount of information on the propagation speeds associated with the problems, in order to accurately detect these steep gradients. This information is then coupled with high-order, non-oscillatory reconstruction of the approximate solution in `the direction of smoothness': that is, information of smoothness does not cross regions of steep gradients. The use of central stencils enable us to realize the reconstructed solutions through simple quadratures. In this manner, central schemes avoid the intricate and time-consuming details of the eigen-structure of the underlying PDEs, and in particular, the use of (approximate) Riemann solvers, dimensional splitting, etc. The resulting family of central schemes offers relatively simple, ``black-box'' solvers for a wide variety of problems governed by multi-dimensional systems of non-linear hyperbolic PDEs.

 

CentPack is a collection of freely distributed C++ routines that implement a number of high-order, non-oscillatory central schemes for hyperbolic systems of conservation laws in one- and two-space dimensions,

u_t + f(u)_x + g(u)_y= 0.

The efficiency and versatility of central schemes resides, mainly, in their simplicity: they eliminate the need for Riemann solvers and avoid dimensional splitting, yielding a generic formulation valid for any hyperbolic system that can be written in the above form. Only information specific to the model and problem to be solved needs to be provided; namely, a description of the flux functions f(u) and g(u), and the appropriate initial and boundary conditions.

The numerical algorithm for the implementation of central schemes consists of two main steps: (i) a non-oscillatory polynomial reconstruction of point values from their cell averages, followed by (ii) the time evolution of the flux functions f(u) and g(u).