Content area
Abstract
Many physical laws can be interpreted as conservation equations (e.g., conservation of mass, momentum, and energy). Furthermore, if the phenomenon described by such equations consists entirely of information propagating at finite speeds, then these conservation equations can be written as hyperbolic conservation laws. Examples of hyperbolic conservation laws include the shallow water equations, which model the dynamics of a thin layer of fluid that is in hydrostatic balance in the vertical direction, and the compressible Euler equations, which model the dynamics of an ideal gas. An important feature of nonlinear hyperbolic conservation laws such as shallow water and compressible Euler is that solution can become discontinuous (i.e., formation of shock waves) in finite time even from smooth initial data. The resulting systems of nonlinear partial differential equations cannot be solved exactly (except for special initial and boundary data); instead, one needs to make use of numerical methods. One important class of such numerical methods is the discontinuous Galerkin (DG) approach. This work focuses on a specific version of the DG method known as the Lax-Wendroff DG (LxW-DG) approach. In particular, this work is concerned with formulating LxW-DG as a prediction/correction scheme, where the prediction step is locally implicit. While high-order versions of the DG methods are accurate for smooth solutions, they can produce unphysical solutions when the underlying solution is discontinuous; in particular, the numerical solution can produce highly oscillatory solutions (Gibbs phenomenon) near discontinuities or even solutions with negative heights (shallow water equations) or negative densities or pressures (compressible Euler equations). Negative heights, densities, or pressures cause the system to lose hyperbolicity and generally cause the numerical methods to go unstable and the simulations to fail. In order to prevent such unphysical solutions, the LxW-DG methodology can be augmented with several types of limiters, or post-processing steps, that are applied during each time step of the method. The challenge is to develop limiters that prevent unphysical solutions while not affecting the method’s high-order accuracy in smooth regimes. In this work, we develop four types of limiters that are applied during various phases of a single-time step of LxW-DG: (1) an element-by-element positivity limiter in the prediction step, (2) a blended Lax-Friedrichs flux limiter that prevents negative element averages, (3) an element-by-element positivity limiter in the correction step, and (4) a final limiter that minimizes unphysical oscillations near discontinuities. Furthermore, we implement the LxW-DG method and the proposed limiting strategy into a freely available Python code for the one-dimensional case and an open-source C++ code called DoGPack for the twodimensional case. The resulting methods are tested on several standard test cases for the shallow water and compressible Euler equations.