  High-order temporal discretizations for hyperbolic conservation laws have
historically been formulated as either a method of lines (MOL) or a
Lax-Wendroff method. In the MOL viewpoint, the partial differential equation is
treated as a large system of ordinary differential equations (ODEs), where an
ODE tailored time-integrator is applied. In contrast, Lax-Wendroff
discretizations immediately convert Taylor series in time to discrete spatial
derivatives. In this work, we propose the Picard integral formulation (PIF),
which is based on the method of modified fluxes, and is used to derive new
Taylor and Runge-Kutta (RK) methods. In particular, we construct a new class of
conservative finite difference methods by applying WENO reconstructions to the
so-called "time-averaged" fluxes. Our schemes are automatically conservative
under any modification of the fluxes, which is attributed to the fact that
classical WENO reconstructions conserve mass when coupled with forward Euler
time steps. The proposed Lax-Wendroff discretization is constructed by taking
Taylor series of the flux function as opposed to Taylor series of the conserved
variables. The RK discretization differs from classical MOL formulations
because we apply WENO reconstructions to time-averaged fluxes rather than
taking linear combinations of spatial derivatives of the flux. In both cases,
we only need one projection onto the characteristic variables per time step.
The PIF is generic, and lends itself to a multitude of options for further
investigation. At present, we present two canonical examples: one based on
Taylor, and the other based on the classical RK method. Stability analyses are
presented for each method. The proposed schemes are applied to hyperbolic
conservation laws in one- and two-dimensions and the results are in good
agreement with current state of the art methods.
