Welcome to Central Station,
a collection of resources on high-resolution central schemes. The
family of central schemes offers "black-box" solvers for a wide
variety of problems governed by multi-dimensional systems of non-linear
conservation laws and related Partial Differential Equations (PDEs).
Along with an extensive archive of publications
focused on the development of central schemes and their application
to a wide range of scientific problems, we present here
CentPack -- a highly adaptable
collection of C++ routines that implement several high-order central
schemes for general hyperbolic systems in one and two space dimensions.
Full documentation for the
download, compilation and execution of
examples are also provided.
read more ...
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
read even more ...
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,
ut + 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).
Jorge Balbás and
April 2010: v1.0.5
Download the pre-compiled
binaries or the
source code of fully-discrete
and semi- discrete central solvers for 1D and 2D systems.
2D scalar examples
2D Euler Equations
2D MHD Equations
Search, download or submit to the
archive of publications dedicated
to the design, analysis and implementation of high-resolution
Click on the images above to see an animation of the interaction
of a high-density MHD cloud with a strong shock. Left: density contours.
Right: magnetic field.