  Nonlinear hyperbolic conservation laws admit singular solutions such as
shockwaves (discontinuities in conserved variables), rarefaction waves
(discontinuities in derivatives), and vacuum states (loss of strong
hyperbolicity). When ostensibly high-order numerical methods are applied in
such solution regimes, unphysical oscillations present themselves that can lead
to large errors and a breakdown of the numerical simulation. In this work we
develop a new Lax-Wendroff discontinuous Galerkin (LxW-DG) method with a
limiting strategy that keeps the solution non-oscillatory and
positivity-preserving for relevant variables, such as height in the shallow
water equations and density and pressure in the compressible Euler equations.
The proposed LxW-DG scheme updates the solution over each time-step with a
locally-implicit predictor followed by an explicit corrector. The
locally-implicit prediction phase is formulated in terms of primitive
variables, which greatly simplifies the solver. The resulting system of
nonlinear algebraic equations are approximately solved via a Picard iteration,
where the number of iterations is equal to the order of accuracy of the method.
The correction phase is an explicit evaluation formulated in terms of
conservative variables in order to guarantee numerical conservation. In order
to achieve full positivity-preservation, limiting is required in both the
prediction and correction steps. The resulting scheme is applied to several
standard test cases for the shallow water and compressible Euler equations. All
of the presented examples are written in a freely available open-source Python
code.
