Limiters::Krivodonova< VolumeDim, tmpl::list< Tags... > > Class Template Reference

An implementation of the Krivodonova limiter. More...

#include <Krivodonova.hpp>

Classes
struct	Alphas
	The \(\alpha_i\) values in the Krivodonova algorithm. More...
struct	DisableForDebugging
	Turn the limiter off. More...
struct	PackagedData

Public Types
using	options = tmpl::list< Alphas, DisableForDebugging >
using	package_argument_tags = tmpl::list< Tags..., domain::Tags::Mesh< VolumeDim > >
using	limit_tags = tmpl::list< Tags... >
using	limit_argument_tags = tmpl::list< domain::Tags::Element< VolumeDim >, domain::Tags::Mesh< VolumeDim > >

Public Member Functions
	Krivodonova (std::array< double, Spectral::maximum_number_of_points< Spectral::Basis::Legendre > > alphas, bool disable_for_debugging=false, const Options::Context &context={})
	Krivodonova (const Krivodonova &)=default
Krivodonova &	operator= (const Krivodonova &)=default
	Krivodonova (Krivodonova &&)=default
Krivodonova &	operator= (Krivodonova &&)=default
void	pup (PUP::er &p)
bool	operator== (const Krivodonova &rhs) const
void	package_data (gsl::not_null< PackagedData * > packaged_data, const typename Tags::type &... tensors, const Mesh< VolumeDim > &mesh, const OrientationMap< VolumeDim > &orientation_map) const
	Package data for sending to neighbor elements.
bool	operator() (const gsl::not_null< std::add_pointer_t< typename Tags::type > >... tensors, const Element< VolumeDim > &element, const Mesh< VolumeDim > &mesh, const std::unordered_map< DirectionalId< VolumeDim >, PackagedData, boost::hash< DirectionalId< VolumeDim > > > &neighbor_data) const
template<typename Tag >
char	limit_one_tensor (const gsl::not_null< Variables< tmpl::list<::Tags::Modal< Tags >... > > * > coeffs_self, const gsl::not_null< bool * > limited_any_component, const Mesh< 1 > &mesh, const std::unordered_map< DirectionalId< 1 >, PackagedData, boost::hash< DirectionalId< 1 > > > &neighbor_data) const
template<typename Tag >
char	limit_one_tensor (const gsl::not_null< Variables< tmpl::list<::Tags::Modal< Tags >... > > * > coeffs_self, const gsl::not_null< bool * > limited_any_component, const Mesh< 2 > &mesh, const std::unordered_map< DirectionalId< 2 >, PackagedData, boost::hash< DirectionalId< 2 > > > &neighbor_data) const
template<typename Tag >
char	limit_one_tensor (const gsl::not_null< Variables< tmpl::list<::Tags::Modal< Tags >... > > * > coeffs_self, const gsl::not_null< bool * > limited_any_component, const Mesh< 3 > &mesh, const std::unordered_map< DirectionalId< 3 >, PackagedData, boost::hash< DirectionalId< 3 > > > &neighbor_data) const
template<typename Tag , size_t Dim>
char	fill_variables_tag_with_spectral_coeffs (const gsl::not_null< Variables< tmpl::list<::Tags::Modal< Tags >... > > * > modal_coeffs, const typename Tag::type &nodal_tensor, const Mesh< Dim > &mesh) const

Static Public Attributes
static constexpr Options::String	help

Detailed Description

template<size_t VolumeDim, typename... Tags>
class Limiters::Krivodonova< VolumeDim, tmpl::list< Tags... > >

An implementation of the Krivodonova limiter.

The limiter is described in [113]. The Krivodonova limiter works by limiting the highest derivatives/modal coefficients using an aggressive minmod approach, decreasing in derivative/modal coefficient order until no more limiting is necessary. In 3d, the function being limited is expanded as:

\begin{align} u^{l,m,n}=\sum_{i,j,k=0,0,0}^{N_i,N_j,N_k}c^{l,m,n}_{i,j,k} P_{i}(\xi)P_{j}(\eta)P_{k}(\zeta) \end{align}

where \(\left\{\xi, \eta, \zeta\right\}\) are the logical coordinates, \(P_{i}\) are the Legendre polynomials, the superscript \(\{l,m,n\}\) represents the element indexed by \(l,m,n\), and \(N_i,N_j\) and \(N_k\) are the number of collocation points minus one in the \(\xi,\eta,\) and \(\zeta\) direction, respectively. The coefficients are limited according to:

\begin{align} \tilde{c}^{l,m,n}_{i,j,k}=\mathrm{minmod} &\left(c_{i,j,k}^{l,m,n}, \alpha_i\left(c^{l+1,m,n}_{i-1,j,k}-c^{l,m,n}_{i-1,j,k}\right), \alpha_i\left(c^{l,m,n}_{i-1,j,k}-c^{l-1,m,n}_{i-1,j,k}\right), \right.\notag \\ &\;\;\;\; \alpha_j\left(c^{l,m+1,n}_{i,j-1,k}-c^{l,m,n}_{i,j-1,k}\right), \alpha_j\left(c^{l,m,n}_{i,j-1,k}-c^{l,m-1,n}_{i,j-1,k}\right), \notag \\ &\;\;\;\;\left. \alpha_k\left(c^{l,m,n+1}_{i,j,k-1}-c^{l,m,n}_{i,j,k-1}\right), \alpha_k\left(c^{l,m,n}_{i,j,k-1}-c^{l,m,n-1}_{i,j,k-1}\right) \right), \label{eq:krivodonova 3d minmod} \end{align}

where \(\mathrm{minmod}\) is the minmod function defined as

\begin{align} \mathrm{minmod}(a,b,c,\ldots)= \left\{ \begin{array}{ll} \mathrm{sgn}(a)\min(\lvert a\rvert, \lvert b\rvert, \lvert c\rvert, \ldots) & \mathrm{if} \; \mathrm{sgn}(a)=\mathrm{sgn}(b)=\mathrm{sgn}(c)=\mathrm{sgn}(\ldots) \\ 0 & \mathrm{otherwise} \end{array}\right. \end{align}

Krivodonova [113] requires \(\alpha_i\) to be in the range

\begin{align*} \frac{1}{2(2i-1)}\le \alpha_i \le 1 \end{align*}

where the lower bound comes from finite differencing the coefficients between neighbor elements when using Legendre polynomials (see [113] for details). Note that we normalize our Legendre polynomials by \(P_i(1) = 1\); this is the normalization [113] uses in 1D, (but not in 2D), which is why our bounds on \(\alpha_i\) match Eq. 14 of [113] (but not Eq. 23). We relax the lower bound:

\begin{align*} 0 \le \alpha_i \le 1 \end{align*}

to allow different basis functions (e.g. Chebyshev polynomials) and to allow the limiter to be more dissipative if necessary. The same \(\alpha_i\)s are used in all dimensions.

Note

The only place where the specific choice of 1d basis comes in is the lower bound for the \(\alpha_i\)s, and so in general the limiter can be applied to any 1d or tensor product of 1d basis functions.

The limiting procedure must be applied from the highest derivatives to the lowest, i.e. the highest coefficients to the lowest. Let us consider a 3d element with \(N+1\) coefficients in each dimension and denote the coefficients as \(c_{i,j,k}\). Then the limiting procedure starts at \(c_{N,N,N}\), followed by \(c_{N,N,N-1}\), \(c_{N,N-1,N}\), and \(c_{N-1,N,N}\). A detailed example is given below. Limiting is stopped if all symmetric pairs of coefficients are left unchanged, i.e. \(c_{i,j,k}=\tilde{c}_{i,j,k}\). By all symmetric coefficients we mean that, for example, \(c_{N-i,N-j,N-k}\), \(c_{N-j,N-i,N-k}\), \(c_{N-k,N-j,N-i}\), \(c_{N-j,N-k,N-i}\), \(c_{N-i,N-k,N-j}\), and \(c_{N-k,N-i,N-j}\) are not limited. As a concrete example, consider a 3d element with 3 collocation points per dimension. Each limited coefficient is defined as (though only computed if needed):

\begin{align*} \tilde{c}^{l,m,n}_{2,2,2}=\mathrm{minmod} &\left(c_{2,2,2}^{l,m,n}, \alpha_2\left(c^{l+1,m,n}_{1,2,2}-c^{l,m,n}_{1,2,2}\right), \alpha_2\left(c^{l,m,n}_{1,2,2}-c^{l-1,m,n}_{1,2,2}\right), \right.\\ &\;\;\;\; \alpha_2\left(c^{l,m+1,n}_{2,1,2}-c^{l,m,n}_{2,1,2}\right), \alpha_2\left(c^{l,m,n}_{2,1,2}-c^{l,m-1,n}_{2,1,2}\right),\\ &\;\;\;\;\left. \alpha_2\left(c^{l,m,n+1}_{2,2,1}-c^{l,m,n}_{2,2,1}\right), \alpha_2\left(c^{l,m,n}_{2,2,1}-c^{l,m,n-1}_{2,2,1}\right) \right),\\ \tilde{c}^{l,m,n}_{2,2,1}=\mathrm{minmod} &\left(c_{2,2,1}^{l,m,n}, \alpha_2\left(c^{l+1,m,n}_{1,2,1}-c^{l,m,n}_{1,2,1}\right), \alpha_2\left(c^{l,m,n}_{1,2,1}-c^{l-1,m,n}_{1,2,1}\right), \right.\\ &\;\;\;\; \alpha_2\left(c^{l,m+1,n}_{2,1,1}-c^{l,m,n}_{2,1,1}\right), \alpha_2\left(c^{l,m,n}_{2,1,1}-c^{l,m-1,n}_{2,1,1}\right),\\ &\;\;\;\;\left. \alpha_1\left(c^{l,m,n+1}_{2,2,0}-c^{l,m,n}_{2,2,0}\right), \alpha_1\left(c^{l,m,n}_{2,2,0}-c^{l,m,n-1}_{2,2,0}\right) \right),\\ \tilde{c}^{l,m,n}_{2,1,2}=\mathrm{minmod} &\left(c_{2,1,2}^{l,m,n}, \alpha_2\left(c^{l+1,m,n}_{1,1,2}-c^{l,m,n}_{1,1,2}\right), \alpha_2\left(c^{l,m,n}_{1,1,2}-c^{l-1,m,n}_{1,1,2}\right), \right.\\ &\;\;\;\; \alpha_1\left(c^{l,m+1,n}_{2,0,2}-c^{l,m,n}_{2,0,2}\right), \alpha_1\left(c^{l,m,n}_{2,0,2}-c^{l,m-1,n}_{2,0,2}\right),\\ &\;\;\;\;\left. \alpha_2\left(c^{l,m,n+1}_{2,1,1}-c^{l,m,n}_{2,1,1}\right), \alpha_2\left(c^{l,m,n}_{2,1,1}-c^{l,m,n-1}_{2,1,1}\right) \right),\\ \tilde{c}^{l,m,n}_{1,2,2}=\mathrm{minmod} &\left(c_{1,2,2}^{l,m,n}, \alpha_1\left(c^{l+1,m,n}_{0,2,2}-c^{l,m,n}_{0,2,2}\right), \alpha_1\left(c^{l,m,n}_{0,2,2}-c^{l-1,m,n}_{0,2,2}\right), \right.\\ &\;\;\;\; \alpha_2\left(c^{l,m+1,n}_{1,1,2}-c^{l,m,n}_{1,1,2}\right), \alpha_2\left(c^{l,m,n}_{1,1,2}-c^{l,m-1,n}_{1,1,2}\right),\\ &\;\;\;\;\left. \alpha_2\left(c^{l,m,n+1}_{1,2,1}-c^{l,m,n}_{1,2,1}\right), \alpha_2\left(c^{l,m,n}_{1,2,1}-c^{l,m,n-1}_{1,2,1}\right) \right),\\ \tilde{c}^{l,m,n}_{2,2,0}=\mathrm{minmod} &\left(c_{2,2,0}^{l,m,n}, \alpha_2\left(c^{l+1,m,n}_{1,2,0}-c^{l,m,n}_{1,2,0}\right), \alpha_2\left(c^{l,m,n}_{1,2,0}-c^{l-1,m,n}_{1,2,0}\right), \right.\\ &\;\;\;\;\left. \alpha_2\left(c^{l,m+1,n}_{2,1,0}-c^{l,m,n}_{2,1,0}\right), \alpha_2\left(c^{l,m,n}_{2,1,0}-c^{l,m-1,n}_{2,1,0}\right) \right),\\ \tilde{c}^{l,m,n}_{2,0,2}=\mathrm{minmod} &\left(c_{2,0,2}^{l,m,n}, \alpha_2\left(c^{l+1,m,n}_{1,0,2}-c^{l,m,n}_{1,0,2}\right), \alpha_2\left(c^{l,m,n}_{1,0,2}-c^{l-1,m,n}_{1,0,2}\right), \right.\\ &\;\;\;\;\left. \alpha_2\left(c^{l,m,n+1}_{2,0,1}-c^{l,m,n}_{2,0,1}\right), \alpha_2\left(c^{l,m,n}_{2,0,1}-c^{l,m,n-1}_{2,0,1}\right) \right),\\ \tilde{c}^{l,m,n}_{0,2,2}=\mathrm{minmod} &\left(c_{0,2,2}^{l,m,n}, \alpha_2\left(c^{l,m+1,n}_{0,1,2}-c^{l,m,n}_{0,1,2}\right), \alpha_2\left(c^{l,m,n}_{0,1,2}-c^{l,m-1,n}_{0,1,2}\right), \right.\\ &\;\;\;\;\left. \alpha_2\left(c^{l,m,n+1}_{0,2,1}-c^{l,m,n}_{0,2,1}\right), \alpha_2\left(c^{l,m,n}_{0,2,1}-c^{l,m,n-1}_{0,2,1}\right) \right),\\ \tilde{c}^{l,m,n}_{2,1,1}=\mathrm{minmod} &\left(c_{2,1,1}^{l,m,n}, \alpha_2\left(c^{l+1,m,n}_{1,1,1}-c^{l,m,n}_{1,1,1}\right), \alpha_2\left(c^{l,m,n}_{1,1,1}-c^{l-1,m,n}_{1,1,1}\right), \right.\\ &\;\;\;\; \alpha_1\left(c^{l,m+1,n}_{2,0,1}-c^{l,m,n}_{2,0,1}\right), \alpha_1\left(c^{l,m,n}_{2,0,1}-c^{l,m-1,n}_{2,0,1}\right),\\ &\;\;\;\;\left. \alpha_1\left(c^{l,m,n+1}_{2,1,0}-c^{l,m,n}_{2,1,0}\right), \alpha_1\left(c^{l,m,n}_{2,1,0}-c^{l,m,n-1}_{2,1,0}\right) \right),\\ \tilde{c}^{l,m,n}_{1,2,1}=\mathrm{minmod} &\left(c_{1,2,1}^{l,m,n}, \alpha_1\left(c^{l+1,m,n}_{0,2,1}-c^{l,m,n}_{0,2,1}\right), \alpha_1\left(c^{l,m,n}_{0,2,1}-c^{l-1,m,n}_{0,2,1}\right), \right.\\ &\;\;\;\; \alpha_2\left(c^{l,m+1,n}_{1,1,1}-c^{l,m,n}_{1,1,1}\right), \alpha_2\left(c^{l,m,n}_{1,1,1}-c^{l,m-1,n}_{1,1,1}\right),\\ &\;\;\;\;\left. \alpha_1\left(c^{l,m,n+1}_{1,2,0}-c^{l,m,n}_{1,2,0}\right), \alpha_1\left(c^{l,m,n}_{1,2,0}-c^{l,m,n-1}_{1,2,0}\right) \right),\\ \tilde{c}^{l,m,n}_{1,1,2}=\mathrm{minmod} &\left(c_{1,1,2}^{l,m,n}, \alpha_1\left(c^{l+1,m,n}_{0,1,2}-c^{l,m,n}_{0,1,2}\right), \alpha_1\left(c^{l,m,n}_{0,1,2}-c^{l-1,m,n}_{0,1,2}\right), \right.\\ &\;\;\;\; \alpha_1\left(c^{l,m+1,n}_{1,0,2}-c^{l,m,n}_{1,0,2}\right), \alpha_1\left(c^{l,m,n}_{1,0,2}-c^{l,m-1,n}_{1,0,2}\right),\\ &\;\;\;\;\left. \alpha_2\left(c^{l,m,n+1}_{1,1,1}-c^{l,m,n}_{1,1,1}\right), \alpha_2\left(c^{l,m,n}_{1,1,1}-c^{l,m,n-1}_{1,1,1}\right) \right), \end{align*}

\begin{align*} \tilde{c}^{l,m,n}_{2,1,0}=\mathrm{minmod} &\left(c_{2,1,0}^{l,m,n}, \alpha_2\left(c^{l+1,m,n}_{1,1,0}-c^{l,m,n}_{1,1,0}\right), \alpha_2\left(c^{l,m,n}_{1,1,0}-c^{l-1,m,n}_{1,1,0}\right), \right.\\ &\;\;\;\;\left. \alpha_1\left(c^{l,m+1,n}_{2,0,0}-c^{l,m,n}_{2,0,0}\right), \alpha_1\left(c^{l,m,n}_{2,0,0}-c^{l,m-1,n}_{2,0,0}\right) \right),\\ \tilde{c}^{l,m,n}_{2,0,1}=\mathrm{minmod} &\left(c_{2,0,1}^{l,m,n}, \alpha_2\left(c^{l+1,m,n}_{1,0,1}-c^{l,m,n}_{1,0,1}\right), \alpha_2\left(c^{l,m,n}_{1,0,1}-c^{l-1,m,n}_{1,0,1}\right), \right.\\ &\;\;\;\;\left. \alpha_1\left(c^{l,m,n+1}_{2,0,0}-c^{l,m,n}_{2,0,0}\right), \alpha_1\left(c^{l,m,n}_{2,0,0}-c^{l,m,n-1}_{2,0,0}\right) \right),\\ \tilde{c}^{l,m,n}_{1,2,0}=\mathrm{minmod} &\left(c_{1,2,0}^{l,m,n}, \alpha_1\left(c^{l+1,m,n}_{0,2,0}-c^{l,m,n}_{0,2,0}\right), \alpha_1\left(c^{l,m,n}_{0,2,0}-c^{l-1,m,n}_{0,2,0}\right), \right.\\ &\;\;\;\;\left. \alpha_2\left(c^{l,m+1,n}_{1,1,0}-c^{l,m,n}_{1,1,0}\right), \alpha_2\left(c^{l,m,n}_{1,1,0}-c^{l,m-1,n}_{1,1,0}\right) \right),\\ \tilde{c}^{l,m,n}_{1,0,2}=\mathrm{minmod} &\left(c_{1,0,2}^{l,m,n}, \alpha_1\left(c^{l+1,m,n}_{0,0,2}-c^{l,m,n}_{0,0,2}\right), \alpha_1\left(c^{l,m,n}_{0,0,2}-c^{l-1,m,n}_{0,0,2}\right), \right.\\ &\;\;\;\;\left. \alpha_2\left(c^{l,m,n+1}_{1,0,1}-c^{l,m,n}_{1,0,1}\right), \alpha_2\left(c^{l,m,n}_{1,0,1}-c^{l,m,n-1}_{1,0,1}\right) \right),\\ \tilde{c}^{l,m,n}_{0,1,2}=\mathrm{minmod} &\left(c_{0,1,2}^{l,m,n}, \alpha_1\left(c^{l,m+1,n}_{0,0,2}-c^{l,m,n}_{0,0,2}\right), \alpha_1\left(c^{l,m,n}_{0,0,2}-c^{l,m-1,n}_{0,0,2}\right), \right. \\ &\;\;\;\;\left. \alpha_2\left(c^{l,m,n+1}_{0,1,1}-c^{l,m,n}_{0,1,1}\right), \alpha_2\left(c^{l,m,n}_{0,1,1}-c^{l,m,n-1}_{0,1,1}\right) \right),\\ \tilde{c}^{l,m,n}_{0,2,1}=\mathrm{minmod} &\left(c_{0,2,1}^{l,m,n}, \alpha_2\left(c^{l,m+1,n}_{0,1,1}-c^{l,m,n}_{0,1,1}\right), \alpha_2\left(c^{l,m,n}_{0,1,1}-c^{l,m-1,n}_{0,1,1}\right), \right.\\ &\;\;\;\;\left. \alpha_1\left(c^{l,m,n+1}_{0,2,0}-c^{l,m,n}_{0,2,0}\right), \alpha_1\left(c^{l,m,n}_{0,2,0}-c^{l,m,n-1}_{0,2,0}\right) \right), \end{align*}

\begin{align*} \tilde{c}^{l,m,n}_{2,0,0}=\mathrm{minmod} &\left(c_{2,0,0}^{l,m,n}, \alpha_2\left(c^{l+1,m,n}_{1,0,0}-c^{l,m,n}_{1,0,0}\right), \alpha_2\left(c^{l,m,n}_{1,0,0}-c^{l-1,m,n}_{1,0,0}\right) \right),\\ \tilde{c}^{l,m,n}_{0,2,0}=\mathrm{minmod} &\left(c_{0,2,0}^{l,m,n}, \alpha_2\left(c^{l,m+1,n}_{0,1,0}-c^{l,m,n}_{0,1,0}\right), \alpha_2\left(c^{l,m,n}_{0,1,0}-c^{l,m-1,n}_{0,1,0}\right) \right),\\ \tilde{c}^{l,m,n}_{0,0,2}=\mathrm{minmod} &\left(c_{0,0,2}^{l,m,n}, \alpha_2\left(c^{l,m,n+1}_{0,0,1}-c^{l,m,n}_{0,0,1}\right), \alpha_2\left(c^{l,m,n}_{0,0,1}-c^{l,m,n-1}_{0,0,1}\right) \right),\\ \tilde{c}^{l,m,n}_{1,1,1}=\mathrm{minmod} &\left(c_{1,1,1}^{l,m,n}, \alpha_1\left(c^{l+1,m,n}_{0,1,1}-c^{l,m,n}_{0,1,1}\right), \alpha_1\left(c^{l,m,n}_{0,1,1}-c^{l-1,m,n}_{0,1,1}\right), \right.\\ &\;\;\;\; \alpha_1\left(c^{l,m+1,n}_{1,0,1}-c^{l,m,n}_{1,0,1}\right), \alpha_1\left(c^{l,m,n}_{1,0,1}-c^{l,m-1,n}_{1,0,1}\right),\\ &\;\;\;\;\left. \alpha_1\left(c^{l,m,n+1}_{1,1,0}-c^{l,m,n}_{1,1,0}\right), \alpha_1\left(c^{l,m,n}_{1,1,0}-c^{l,m,n-1}_{1,1,0}\right) \right),\\ \tilde{c}^{l,m,n}_{1,1,0}=\mathrm{minmod} &\left(c_{1,1,0}^{l,m,n}, \alpha_1\left(c^{l+1,m,n}_{0,1,0}-c^{l,m,n}_{0,1,0}\right), \alpha_1\left(c^{l,m,n}_{0,1,0}-c^{l-1,m,n}_{0,1,0}\right), \right.\\ &\;\;\;\;\left. \alpha_1\left(c^{l,m+1,n}_{1,0,0}-c^{l,m,n}_{1,0,0}\right), \alpha_1\left(c^{l,m,n}_{1,0,0}-c^{l,m-1,n}_{1,0,0}\right), \right),\\ \tilde{c}^{l,m,n}_{1,0,1}=\mathrm{minmod} &\left(c_{1,0,1}^{l,m,n}, \alpha_1\left(c^{l+1,m,n}_{0,0,1}-c^{l,m,n}_{0,0,1}\right), \alpha_1\left(c^{l,m,n}_{0,0,1}-c^{l-1,m,n}_{0,0,1}\right), \right.\\ &\;\;\;\;\left. \alpha_1\left(c^{l,m,n+1}_{1,0,0}-c^{l,m,n}_{1,0,0}\right), \alpha_1\left(c^{l,m,n}_{1,0,0}-c^{l,m,n-1}_{1,0,0}\right) \right),\\ \tilde{c}^{l,m,n}_{0,1,1}=\mathrm{minmod} &\left(c_{0,1,1}^{l,m,n}, \alpha_1\left(c^{l,m+1,n}_{0,0,1}-c^{l,m,n}_{0,0,1}\right), \alpha_1\left(c^{l,m,n}_{0,0,1}-c^{l,m-1,n}_{0,0,1}\right), \right.\\ &\;\;\;\;\left. \alpha_1\left(c^{l,m,n+1}_{0,1,0}-c^{l,m,n}_{0,1,0}\right), \alpha_1\left(c^{l,m,n}_{0,1,0}-c^{l,m,n-1}_{0,1,0}\right) \right),\\ \tilde{c}^{l,m,n}_{1,0,0}=\mathrm{minmod} &\left(c_{1,0,0}^{l,m,n}, \alpha_1\left(c^{l+1,m,n}_{0,0,0}-c^{l,m,n}_{0,0,0}\right), \alpha_1\left(c^{l,m,n}_{0,0,0}-c^{l-1,m,n}_{0,0,0}\right) \right),\\ \tilde{c}^{l,m,n}_{0,1,0}=\mathrm{minmod} &\left(c_{0,1,0}^{l,m,n}, \alpha_1\left(c^{l,m+1,n}_{0,0,0}-c^{l,m,n}_{0,0,0}\right), \alpha_1\left(c^{l,m,n}_{0,0,0}-c^{l,m-1,n}_{0,0,0}\right) \right),\\ \tilde{c}^{l,m,n}_{0,0,1}=\mathrm{minmod} &\left(c_{0,0,1}^{l,m,n}, \alpha_1\left(c^{l,m,n+1}_{0,0,0}-c^{l,m,n}_{0,0,0}\right), \alpha_1\left(c^{l,m,n}_{0,0,0}-c^{l,m,n-1}_{0,0,0}\right) \right), \end{align*}

The algorithm to perform the limiting is as follows:

• limit \(c_{2,2,2}\) (i.e. \(c_{2,2,2}\leftarrow\tilde{c}_{2,2,2}\)), if not changed, stop
• limit \(c_{2,2,1}\), \(c_{2,1,2}\), and \(c_{1,2,2}\), if all not changed, stop
• limit \(c_{2,2,0}\), \(c_{2,0,2}\), and \(c_{0,2,2}\), if all not changed, stop
• limit \(c_{2,1,1}\), \(c_{1,2,1}\), and \(c_{1,1,2}\), if all not changed, stop
• limit \(c_{2,1,0}\), \(c_{2,0,1}\), \(c_{1,2,0}\), \(c_{1,0,2}\), \(c_{0,1,2}\), and \(c_{0,2,1}\), if all not changed, stop
• limit \(c_{2,0,0}\), \(c_{0,2,0}\), and \(c_{0,0,2}\), if all not changed, stop
• limit \(c_{1,1,1}\), if not changed, stop
• limit \(c_{1,1,0}\), \(c_{1,0,1}\), and \(c_{0,1,1}\), if all not changed, stop
• limit \(c_{1,0,0}\), \(c_{0,1,0}\), and \(c_{0,0,1}\), if all not changed, stop

The 1d and 2d implementations are straightforward restrictions of the described algorithm.

Limitations:

• We currently recompute the spectral coefficients for local use, after having computed these same coefficients for sending to the neighbors. We should be able to avoid this either by storing the coefficients in the DataBox or by allowing the limiters' packaged_data function to return an object to be passed as an additional argument to the operator() (still would need to be stored in the DataBox).
• We cannot handle the case where neighbors have more/fewer coefficients. In the case that the neighbor has more coefficients we could just ignore the higher coefficients. In the case that the neighbor has fewer coefficients we have a few choices.
• Having a different number of collocation points in different directions is not supported. However, it is straightforward to handle this case. The outermost loop should be in the direction with the most collocation points, while the inner most loop should be over the direction with the fewest collocation points. The highest to lowest coefficients can then be limited appropriately again.
• h-refinement is not supported, but there is one reasonably straightforward implementation that may work. In this case we would ignore refinement that is not in the direction of the neighbor, treating the element as simply having multiple neighbors in that direction. The only change would be accounting for different refinement in the direction of the neighor, which should be easy to add since the differences in coefficients in Eq. \(\ref{eq:krivodonova 3d minmod}\) will just be multiplied by non-unity factors.

Member Data Documentation

help

template<size_t VolumeDim, typename... Tags>

constexpr Options::String Limiters::Krivodonova< VolumeDim, tmpl::list< Tags... > >::help

staticconstexpr

Initial value:

= {
      "The hierarchical limiter of Krivodonova.\n\n"
      "This limiter works by limiting the highest modal "
      "coefficients/derivatives using an aggressive minmod approach, "
      "decreasing in modal coefficient order until no more limiting is "
      "necessary."}

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The documentation for this class was generated from the following file:

• src/Evolution/DiscontinuousGalerkin/Limiters/Krivodonova.hpp