  The goal of this work is to introduce new families of shock-capturing
high-order numerical methods for systems of conservation laws that combine Fast
WENO (FWENO) and Optimal WENO (OWENO) reconstructions with Approximate Taylor
methods for the time discretization. FWENO reconstructions are based on
smoothness indicators that require a lower number of calculations than the
standard ones. OWENO reconstructions are based on a definition of the nonlinear
weights that allows one to unconditionally attain the optimal order of accuracy
regardless of the order of critical points. Approximate Taylor methods update
the numerical solutions by using a Taylor expansion in time in which, instead
of using the Cauchy-Kovalevskaya procedure, the time derivatives are computed
by combining spatial and temporal numerical differentiation with Taylor
expansions in a recursive way. These new methods are compared between them and
against methods based on standard WENO implementations and/or SSP-RK time
discretization. A number of test cases are considered ranging from scalar
linear 1d problems to nonlinear systems of conservation laws in 2d.
