Along the way, we explain the differences and similarities between optimal prediction, the representer method in data assimilation, and duality methods for finding weak solutions. We also discuss the conditions under which a simple implementation of the optimal prediction method can be expected to perform well. We present a new third-order central scheme for approximating solutions of systems of conservation laws in one and two space dimensions. The heart of the matter is that one of these interpolants is taken as an arbitrary quadratic polynomial and the weights of the convex combination are set as to obtain third-order accuracy in smooth regions. In the one-dimensional case, our new third order scheme is based on an extremely compact four stencil.
Analogous compactness is retained in more space dimensions.
The accuracy, robustness and high-resolution properties of our scheme are demonstrated in a variety of one and two dimensional problems.