2016/01/29 by Scott Moe, James A. Rossmanith, Moe, Scott A. +3 · 2 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1601.08145
openalex publication_date 2016/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work introduces a single-stage, single-step method for the compressible\nEuler equations that is provably positivity-preserving and can be applied on\nboth Cartesian and unstructured meshes. This method is the first case of a\nsingle-stage, single-step method that is simultaneously high-order,\npositivity-preserving, and operates on unstructured meshes. Time-stepping is\naccomplished via the Lax-Wendroff approach, which is also sometimes called the\nCauchy-Kovalevskaya procedure, where temporal derivatives in a Taylor series in\ntime are exchanged for spatial derivatives. The Lax-Wendroff discontinuous\nGalerkin (LxW-DG) method developed in this work is formulated so that it looks\nlike a forward Euler update but with a high-order time-extrapolated flux. In\nparticular, the numerical flux used in this work is a linear combination of a\nlow-order positivity-preserving contribution and a high-order component that\ncan be damped to enforce positivity of the cell averages for the density and\npressure for each time step. In addition to this flux limiter, a moment limiter\nis applied that forces positivity of the solution at finitely many quadrature\npoints within each cell. The combination of the flux limiter and the moment\nlimiter guarantees positivity of the cell averages from one time-step to the\nnext. Finally, a simple shock capturing limiter that uses the same basic\ntechnology as the moment limiter is introduced in order to obtain\nnon-oscillatory results. The resulting scheme can be extended to arbitrary\norder without increasing the size of the effective stencil. We present\nnumerical results in one and two space dimensions that demonstrate the\nrobustness of the proposed scheme.\n