A compact-stencil WENO limiter for DG. More...
A compact-stencil WENO limiter for DG.
Implements the simple WENO limiter of [210] and the Hermite WENO (HWENO) limiter of [211]. The implementation is system-agnostic and can act on an arbitrary set of tensors.
The compact-stencil WENO limiters require communication only between nearest-neighbor elements, but aim to preserve the full order of the DG solution when the solution is smooth. To achieve this, full volume data is communicated between neighbors. For each tensor component to limit, the new solution is obtained by a standard WENO procedure — the new solution is a linear combination of different polynomials, with weights chosen so that the smoother (i.e., less oscillatory) polynomials contribute the most to the sum.
For the simple WENO and HWENO limiters, the polynomials used are the local DG solution as well as a "modified" solution from each neighbor element. For the simple WENO limiter, the modified solution is obtained by simply extrapolating the neighbor solution onto the troubled element. For the HWENO limiter, the modified solution is obtained by a least-squares fit to the solution across multiple neighboring elements.
There are a few differences between the limiters as implemented in SpECTRE and as presented in the references. We list them here and discuss them further below.
Finally, in 4., we will discuss the geometric limitations of the implementation (which are not a deviation from the references).
SpECTRE uses a Legendre basis, rather than the polynomial basis that we understand to be used in the references. Because the construction of the modified neighbor solutions and the WENO sum is geometrically motivated, the overall algorithm should work similarly. However, the precise numerics may differ.
This implementation can act on an arbitrary set of tensors. To reach this generality, our HWENO implementation uses a different troubled-cell indicator (TCI) than the reference, which instead specializes the TCI to the Newtonian Euler system of equations.
This implementation uses the minmod-based TVB TCI of [41] to identify elements that need limiting. The simple WENO implementation follows its reference: it checks the TCI independently for each tensor component, so that only certain tensor components may be limited. The HWENO implementation checks the TVB TCI for all tensor components, and if any single component is troubled, then all components of all tensors are limited.
When the evolution system has multiple evolved variables, the recommendation of the references is to apply the limiter to the system's characteristic variables to reduce spurious post-limiting oscillations. In SpECTRE, applying the limiter to the characteristic variables requires specializing the limiter to each evolution system. The system-specific limiter can also implement a system-specific TCI (as the HWENO reference does) to more precisely trigger the limiter.
We use the oscillation indicator of [61], modified for use on the square/cube grids of SpECTRE. We favor this indicator because portions of the work can be precomputed, leading to an oscillation measure that is efficient to evaluate.
Does not support non-Legendre bases; this is checked in DEBUG mode. In principle other bases could be supported, but this would require generalizing the many internal algorithms that assume a Legendre basis.
Does not support h- or p-refinement; this is checked always. In principle this could be supported. The modified neighbor solution algorithm would need to be generalized (reasonable for simple WENO, extremely tedious for HWENO), and the sum of neighbor solutions may need to be updated as well.
Does not support curved elements; this is not enforced. The code will run but we make no guarantees about the results. Specifically, the limiter acts in the Frame::ElementLogical coordinates, because in these coordinates it is straightforward to formulate the algorithm. This means the limiter can operate on generic deformed grids — however, some things can start to break down, especially on strongly deformed grids:
Frame::ElementLogical to Frame::Inertial) varies across the element, then the limiter fails to be conservative. This is because the integral of a tensor u over the element will change after the limiter activates on u.