## Dissipation

Erik Schnetter <schnetter@aei.mpg.de>, Bernard Kelly <bernard.j.kelly@nasa.gov>

$$Date$$

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,
${\stackrel{\u0303}{\gamma}}_{ij}$ is
treated as an independent ﬁeld for dissipation purposes.

$$\begin{array}{rcll}{\partial}_{t}U& =& {\partial}_{t}U+{\left(-1\right)}^{\left(p+3\right)\u22152}\mathit{\epsilon}\frac{1}{{2}^{p+1}}\left({h}_{x}^{p}\frac{{\partial}^{\left(p+1\right)}}{\partial {x}^{\left(p+1\right)}}+{h}_{y}^{p}\frac{{\partial}^{\left(p+1\right)}}{\partial {y}^{\left(p+1\right)}}+{h}_{z}^{p}\frac{{\partial}^{\left(p+1\right)}}{\partial {z}^{\left(p+1\right)}}\right)U,& \text{}\\ & =& {\partial}_{t}U+{\left(-1\right)}^{\left(p+3\right)\u22152}\mathit{\epsilon}\frac{{h}^{p}}{{2}^{p+1}}\left(\frac{{\partial}^{\left(p+1\right)}}{\partial {x}^{\left(p+1\right)}}+\frac{{\partial}^{\left(p+1\right)}}{\partial {y}^{\left(p+1\right)}}+\frac{{\partial}^{\left(p+1\right)}}{\partial {z}^{\left(p+1\right)}}\right)U,& \text{}\end{array}$$
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 $\mathit{\epsilon}$.

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 $\left(p+1\right)\u22152$
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