## Dissipation

Date

Abstract

Add $n$th-order Kreiss-Oliger dissipation to the right hand side of evolution equations. This thorn is intended for time evolutions that use MoL.

### 1 Physical System

For a description of Kreiss-Oliger artiﬁcial dissipation, see [1].

The additional dissipation terms appear as follows, for a general grid function $U$. Here, the tensor character of the ﬁeld is irrelevant: each component of, say, ${\stackrel{̃}{\gamma }}_{ij}$ is treated as an independent ﬁeld for dissipation purposes.

$\begin{array}{rcll}{\partial }_{t}U& =& {\partial }_{t}U+{\left(-1\right)}^{\left(p+3\right)∕2}𝜖\frac{1}{{2}^{p+1}}\left({h}_{x}^{p}\frac{{\partial }^{\left(p+1\right)}}{\partial {x}^{\left(p+1\right)}}+{h}_{y}^{p}\frac{{\partial }^{\left(p+1\right)}}{\partial {y}^{\left(p+1\right)}}+{h}_{z}^{p}\frac{{\partial }^{\left(p+1\right)}}{\partial {z}^{\left(p+1\right)}}\right)U,& \text{}\\ & =& {\partial }_{t}U+{\left(-1\right)}^{\left(p+3\right)∕2}𝜖\frac{{h}^{p}}{{2}^{p+1}}\left(\frac{{\partial }^{\left(p+1\right)}}{\partial {x}^{\left(p+1\right)}}+\frac{{\partial }^{\left(p+1\right)}}{\partial {y}^{\left(p+1\right)}}+\frac{{\partial }^{\left(p+1\right)}}{\partial {z}^{\left(p+1\right)}}\right)U,& \text{}\end{array}$

where ${h}_{x}$, ${h}_{y}$, and ${h}_{z}$ are the local grid spacings in each Cartesian direction, and the second equality holds in the usual situation where the three are equal: ${h}_{x}={h}_{y}={h}_{z}=h$.

### 2 Implementation in Cactus

The Dissipation thorn’s dissipation rate is controlled by a small number of parameters:

• order is the order $p$ of the dissipation, implying the use of the $\left(p+1\right)$-st spatial derivatives;
• epsdiss is the overall dissipation strength $𝜖$.

Currently available values of order are $p\in \left\{1,3,5,7,9\right\}$. To apply dissipation at order $p$ requires that we have at least $\left(p+1\right)∕2$ ghostzones — $\left\{1,2,3,4,5\right\}$, respectively.

The list of ﬁelds to be dissipated is speciﬁed in the parameter vars. The thorn does not allow