  We propose a class of weighted compact central (WCC) schemes for solving
hyperbolic conservation laws. The linear version can be considered as a
high-order extension of the central Lax-Friedrichs (LxF) scheme and the central
conservation element and solution element (CESE) scheme. On every cell, the
solution is approximated by a Pth order polynomial of which all the DOFs are
stored and updated separately. The cell average is updated by a classical
finite volume scheme which is constructed based on space-time staggered meshes
such that the fluxes are continuous across the interfaces of the adjacent
control volumes and, therefore, the local Riemann problem is bypassed. The kth
order spatial derivatives are updated by a central difference of (k-1)th order
spatial derivatives at cell vertices. All the space-time information is
calculated by the Cauchy-Kovalewski procedure. By doing so, the schemes are
able to achieve arbitrarily uniform spacetime high order on a super-compact
stencil with only one explicit time step. In order to capture discontinuities
without spurious oscillations, a weighted essentially non-oscillatory (WENO)
type limiter is tailor-made for the schemes. The limiter preserves the
compactness and high order accuracy of the schemes. The accuracy, robustness,
and efficiency of the schemes are verified by several numerical examples of
scalar conservation laws and the compressible Euler equations.
