1 research outputs found
Positivity-preserving discontinuous Galerkin methods with Lax-Wendroff time discretizations
This work introduces a single-stage, single-step method for the compressible
Euler equations that is provably positivity-preserving and can be applied on
both Cartesian and unstructured meshes. This method is the first case of a
single-stage, single-step method that is simultaneously high-order,
positivity-preserving, and operates on unstructured meshes. Time-stepping is
accomplished via the Lax-Wendroff approach, which is also sometimes called the
Cauchy-Kovalevskaya procedure, where temporal derivatives in a Taylor series in
time are exchanged for spatial derivatives. The Lax-Wendroff discontinuous
Galerkin (LxW-DG) method developed in this work is formulated so that it looks
like a forward Euler update but with a high-order time-extrapolated flux. In
particular, the numerical flux used in this work is a linear combination of a
low-order positivity-preserving contribution and a high-order component that
can be damped to enforce positivity of the cell averages for the density and
pressure for each time step. In addition to this flux limiter, a moment limiter
is applied that forces positivity of the solution at finitely many quadrature
points within each cell. The combination of the flux limiter and the moment
limiter guarantees positivity of the cell averages from one time-step to the
next. Finally, a simple shock capturing limiter that uses the same basic
technology as the moment limiter is introduced in order to obtain
non-oscillatory results. The resulting scheme can be extended to arbitrary
order without increasing the size of the effective stencil. We present
numerical results in one and two space dimensions that demonstrate the
robustness of the proposed scheme.Comment: 28 pages, 9 figure