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Nonlinear hyperbolic conservation laws admit singular solutions such asshockwaves (discontinuities in conserved variables), rarefaction waves(discontinuities in derivatives), and vacuum states (loss of stronghyperbolicity). When ostensibly high-order numerical methods are applied insuch solution regimes, unphysical oscillations present themselves that can leadto large errors and a breakdown of the numerical simulation. In this work wedevelop a new Lax-Wendroff discontinuous Galerkin (LxW-DG) method with alimiting strategy that keeps the solution non-oscillatory andpositivity-preserving for relevant variables, such as height in the shallowwater equations and density and pressure in the compressible Euler equations.The proposed LxW-DG scheme updates the solution over each time-step with alocally-implicit predictor followed by an explicit corrector. Thelocally-implicit prediction phase is formulated in terms of primitivevariables, which greatly simplifies the solver. The resulting system ofnonlinear 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 ofconservative variables in order to guarantee numerical conservation. In orderto achieve full positivity-preservation, limiting is required in both theprediction and correction steps. The resulting scheme is applied to severalstandard test cases for the shallow water and compressible Euler equations. Allof the presented examples are written in a freely available open-source Pythoncode.

A Positivity-Preserving Limiting Strategy for Locally-Implicit Lax-Wendroff Discontinuous Galerkin Methods