  This paper presents a new approach, so-called boundary variation diminishing
(BVD), for reconstructions that minimize the discontinuities (jumps) at cell
interfaces in Godunov type schemes. It is motivated by the observation that
diminishing the jump at the cell boundary might effectively reduce the
dissipation in numerical flux. Different from the existing practices which seek
high-order polynomials within mesh cells while assuming discontinuities being
always at the cell interfaces, we proposed a new strategy that combines a
high-order polynomial-based interpolation and a jump-like reconstruction that
allows a discontinuity being partly represented within the mesh cell rather
than at the interface. It is shown that new schemes of high fidelity for both
continuous and discontinuous solutions can be devised by the BVD guideline with
properly-chosen candidate reconstruction schemes. Excellent numerical results
have been obtained for both scalar and Euler conservation laws with
substantially improved solution quality in comparison with the existing
methods. This work provides a simple and accurate alternative of great
practical significance to the current Godunov paradigm which overly pursues the
smoothness within mesh cell under the questionable premiss that discontinuities
only appear at cell interfaces.
