## A Multi-Patch Wave Toy

March 31, 2019

### 1 Physical System

The massless wave equation for a scalar ﬁeld $\varphi$ can be written as

${\partial }_{\mu }\left({\gamma }^{\mu \nu }{d}_{\nu }\right)=0$

where ${\gamma }^{\mu \nu }=\sqrt{-g}{g}^{\mu \nu }$, and ${d}_{\mu }\equiv {\partial }_{\mu }\varphi$. Using the expressions $\sqrt{-g}=\alpha \sqrt{h}$ and

${g}^{\mu \nu }=\left(\begin{array}{cc}\hfill -1∕{\alpha }^{2}\hfill & \hfill {\beta }^{i}∕{\alpha }^{2}\hfill \\ \hfill {\beta }^{j}∕{\alpha }^{2}\hfill & \hfill {\gamma }^{ij}-{\beta }^{i}{\beta }^{j}∕{\alpha }^{2}\hfill \end{array}\right),$

where ${\gamma }^{ij}$ is the inverse of the three metric, the equation can be rewritten as

$\begin{array}{rcll}\stackrel{̇}{\varphi }& =& \Pi ,& \text{(1)}\text{}\text{}\\ \stackrel{̇}{\Pi }& =& {\beta }^{i}{\partial }_{i}\Pi +\frac{\alpha }{\sqrt{h}}{\partial }_{i}\left(\frac{\sqrt{h}}{\alpha }{\beta }^{i}\Pi +\frac{\sqrt{h}}{\alpha }{H}^{ij}{d}_{j}\right)+\frac{\alpha }{\sqrt{h}}{d}_{i}{\partial }_{t}\left(\frac{\sqrt{h}{\beta }^{i}}{\alpha }\right)-\frac{\Pi \alpha }{\sqrt{h}}{\partial }_{t}\left(\frac{\sqrt{h}}{\alpha }\right),& \text{(2)}\text{}\text{}\\ \stackrel{̇}{{d}_{i}}& =& {\partial }_{i}\Pi & \text{(3)}\text{}\text{}\end{array}$

with ${H}^{ij}\equiv {\alpha }^{2}{\gamma }^{ij}-{\beta }^{i}{\beta }^{j}$. The non-shift speed modes with respect to a boundary with normal ${n}_{i}$ are

${v}^{±}=\lambda \Pi +{H}^{ij}{n}_{i}{d}_{j}$

and the shift speed modes are ${d}_{A}$, with $A$ transversal directions.

The physical energy is

$E=\frac{1}{2}\int \frac{1}{\alpha }\left[{\Pi }^{2}+{H}^{ij}{d}_{i}{d}_{j}\right]\sqrt{h}d{x}^{3}$

and the way the equations above have been written this energy is not increasing in the stationary background case if homogeneous boundary conditions are given at outer boundaries, also at the discrete level (replacing above ${\partial }_{i}$ by ${D}_{i}$).

The implementation of the ﬁeld equations in the code diﬀers slightly from the above. Assuming a stationary background and expanding out the derivative operator, we obtain the equations in the ﬁnal form

$\begin{array}{rcll}\stackrel{̇}{\varphi }& =& \Pi ,& \text{(4)}\text{}\text{}\\ \stackrel{̇}{\Pi }& =& {\beta }^{i}{\partial }_{i}\Pi +\frac{\alpha }{\sqrt{h}}{\partial }_{i}\left(\frac{\sqrt{h}}{\alpha }{\beta }^{i}\Pi \right)+\frac{\alpha }{\sqrt{h}}{\partial }_{i}\left(\frac{\sqrt{h}}{\alpha }{H}^{ij}\right){\partial }_{j}\varphi +\frac{\alpha }{\sqrt{h}}\frac{\sqrt{h}}{\alpha }{H}^{ij}{\partial }_{i}{\partial }_{j}\varphi .& \text{(5)}\text{}\text{}\end{array}$

### 2 Non-linear addition to the multi-patch wave toy

Simple addition to the wave multi-patch toy to get started on this.

#### 2.1 Physical System

This is just a non-linear wave equation obtained, a very minor modiﬁcation to the wave equation thorn adding a diﬀerent initial data and a slight modiﬁcation to the right hand side. A reference to this is in a paper by Liebling to appear in Phys. Rev. D (2005). The non-linear wave equation is written as

${\partial }_{u}\left({\gamma }^{\mu \nu }{d}_{\nu }\right)={\varphi }^{p}$

where ${\gamma }^{\mu \nu }=\sqrt{-g}{g}^{\mu \nu }$, and ${d}_{\mu }={\partial }_{\mu }\varphi$ and $p$ must be an odd integer $\ge 3$.

The initial data coded is given by

$\begin{array}{rcll}\varphi & =& A{e}^{-{\left({r}_{1}-R\right)}^{2}∕{\delta }^{2}}& \text{(6)}\text{}\text{}\\ \Pi & =& \mu {\varphi }_{,r}+\Omega \left(y{\varphi }_{,x}-x{\varphi }_{,y}\right)& \text{(7)}\text{}\text{}\\ {d}_{i}={\varphi }_{,i}& & & \text{(8)}\text{}\text{}\end{array}$

with ${\stackrel{̃}{r}}^{2}={𝜖}_{x}{x}^{2}+{𝜖}_{y}{y}^{2}+{z}^{2}$.

The parameters used for this initial data are given some distinct names to avoid conﬂicts with existing ones and are as follows:

• initial-data = GaussianNonLinear (choose the above mentioned initial data)
• nonlinearrhs = turn on/oﬀ the right hand side for testing. a boolean variable.
• powerrhs = power $p$ above.
• epsx = ${𝜖}_{x}$
• epsy = ${𝜖}_{y}$
• ANL = $A$
• deltaNL = $\delta$
• omeNL = $\Omega$
• RNL = R

Note, as the solution is not known, one must set for now the incoming ﬁelds to $0$. CPBC might one day be put... though who knows :-)

### 3 Formulations

$\begin{array}{rcll}U& =& {\left[\begin{array}{cccc}\hfill \rho \hfill & \hfill {v}_{x}\hfill & \hfill {v}_{y}\hfill & \hfill {v}_{z}\hfill \end{array}\right]}^{T}={\left[\begin{array}{cc}\hfill \rho \hfill & \hfill {v}_{i}\hfill \end{array}\right]}^{T}& \text{(9)}\text{}\text{}\\ {\partial }_{t}U& =& {A}^{i}{\partial }_{i}U+\cdots & \text{(10)}\text{}\text{}\\ ||{n}_{i}||& \ne & 1& \text{(11)}\text{}\text{}\end{array}$

#### 3.1 $dt$

Setting $\rho ={\partial }_{t}u$.

RHS:

$\begin{array}{rcll}{H}^{ij}& =& {\alpha }^{2}{\gamma }^{ij}-{\beta }^{i}{\beta }^{j}& \text{(12)}\text{}\text{}\\ {\partial }_{t}u& =& \rho & \text{(13)}\text{}\text{}\\ {\partial }_{t}\rho & =& {\beta }^{i}{\partial }_{i}\rho +\frac{\alpha }{𝜖}{\partial }_{i}\frac{𝜖}{\alpha }\left({\beta }^{i}\rho +{H}^{ij}{v}_{j}\right)& \text{(14)}\text{}\text{}\\ {\partial }_{t}{v}_{i}& =& {\partial }_{i}\rho & \text{(15)}\text{}\text{}\end{array}$

Propagation matrix:

$\begin{array}{rcll}{A}^{x}& =& \left(\begin{array}{cccc}\hfill 2{\beta }^{x}\hfill & \hfill -{\beta }^{x}{\beta }^{x}+{\alpha }^{2}{\gamma }^{xx}\hfill & \hfill -{\beta }^{x}{\beta }^{y}+{\gamma }^{xy}{\alpha }^{2}\hfill & \hfill -{\beta }^{x}{\beta }^{z}+{\alpha }^{2}{\gamma }^{xz}\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)& \text{(16)}\text{}\text{}\end{array}$

$\begin{array}{rcll}{A}^{n}& =& \left(\begin{array}{cc}\hfill 2{\beta }^{i}{n}_{i}\hfill & \hfill \left(-{\beta }^{i}{\beta }^{j}+{\alpha }^{2}{\gamma }^{ij}\right){n}_{i}\hfill \\ \hfill {n}_{i}\hfill & \hfill 0\hfill \end{array}\right)& \text{(17)}\text{}\text{}\end{array}$

Eigensystem:

$\begin{array}{rcll}{\lambda }_{1}=0& ,& {w}_{1}={\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]}^{T}& \text{(18)}\text{}\text{}\\ {\lambda }_{2}=0& ,& {w}_{2}={\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]}^{T}& \text{(19)}\text{}\text{}\\ {\lambda }_{3}={\beta }^{x}-\alpha \sqrt{{\gamma }^{xx}}& ,& {w}_{3}={\left[\begin{array}{cccc}\hfill {\beta }^{x}-\alpha \sqrt{{\gamma }^{xx}}\hfill & \hfill {\beta }^{x}{\beta }^{x}+{\alpha }^{2}{\gamma }^{xx}\hfill & \hfill {\beta }^{x}{\beta }^{y}+{\alpha }^{2}{\gamma }^{xy}\hfill & \hfill {\beta }^{x}{\beta }^{z}+{\alpha }^{2}{\gamma }^{xz}\hfill \end{array}\right]}^{T}& \text{(20)}\text{}\text{}\\ {\lambda }_{4}={\beta }^{x}+\alpha \sqrt{{\gamma }^{xx}}& ,& {w}_{4}={\left[\begin{array}{cccc}\hfill {\beta }^{x}+\alpha \sqrt{{\gamma }^{xx}}\hfill & \hfill {\beta }^{x}{\beta }^{x}+{\alpha }^{2}{\gamma }^{xx}\hfill & \hfill {\beta }^{x}{\beta }^{y}+{\alpha }^{2}{\gamma }^{xy}\hfill & \hfill {\beta }^{x}{\beta }^{z}+{\alpha }^{2}{\gamma }^{xz}\hfill \end{array}\right]}^{T}& \text{(21)}\text{}\text{}\end{array}$

$\begin{array}{rcll}{\lambda }_{t}=0& ,& {w}_{t}={\left[\begin{array}{cc}\hfill 0\hfill & \hfill {t}_{i}\hfill \end{array}\right]}^{T}& \text{(22)}\text{}\text{}\\ {\lambda }_{±}={\beta }^{i}{n}_{i}±\alpha \sqrt{{\gamma }^{ij}{n}_{i}{n}_{j}}& ,& {w}_{±}={\left[\begin{array}{cc}\hfill {\beta }^{i}{n}_{i}±\alpha \sqrt{{\gamma }^{ij}{n}_{i}{n}_{j}}\hfill & \hfill {H}^{ij}{n}_{j}\hfill \end{array}\right]}^{T}& \text{(23)}\text{}\text{}\end{array}$

#### 3.2 $d0$

Setting $\rho ={\mathsc{ℒ}}_{n}u$ with ${n}_{a}={D}_{a}t$, leading to $\rho ={\partial }_{0}u=\left(1∕\alpha \right){\partial }_{t}u-\left(1∕\alpha \right){\beta }^{i}{\partial }_{i}u$.

RHS:

$\begin{array}{rcll}{H}^{ij}& =& {\gamma }^{ij}& \text{(24)}\text{}\text{}\\ {\partial }_{t}u& =& \alpha \rho +{\beta }^{i}{v}_{i}& \text{(25)}\text{}\text{}\\ {\partial }_{t}\rho & =& {\beta }^{i}{\partial }_{i}\rho +\frac{1}{𝜖}{\partial }_{i}𝜖\alpha {H}^{ij}{v}_{j}+\frac{\rho }{𝜖}{\partial }_{i}𝜖{\beta }^{i}& \text{(26)}\text{}\text{}\\ {\partial }_{t}{v}_{i}& =& {\beta }^{j}{\partial }_{j}{v}_{i}+{v}_{j}{\partial }_{i}{\beta }^{j}+{\partial }_{i}\alpha \rho & \text{(27)}\text{}\text{}\end{array}$

Propagation matrix:

$\begin{array}{rcll}{A}^{x}& =& \left(\begin{array}{cccc}\hfill {\beta }^{x}\hfill & \hfill \alpha {\gamma }^{xx}\hfill & \hfill \alpha {\gamma }^{xy}\hfill & \hfill \alpha {\gamma }^{xz}\hfill \\ \hfill \alpha \hfill & \hfill {\beta }^{x}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }^{x}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill {\beta }^{x}\hfill \end{array}\right)& \text{(28)}\text{}\text{}\end{array}$

$\begin{array}{rcll}{A}^{n}& =& \left(\begin{array}{cc}\hfill {\beta }^{i}{n}_{i}\hfill & \hfill \alpha {\gamma }^{ij}{n}_{i}\hfill \\ \hfill \alpha {n}_{i}\hfill & \hfill {\beta }^{k}{n}_{k}{\delta }_{ij}\hfill \end{array}\right)& \text{(29)}\text{}\text{}\end{array}$

Eigensystem:

$\begin{array}{rcll}{\lambda }_{t}={\beta }^{i}{n}_{i}& ,& {w}_{t}={\left[\begin{array}{cc}\hfill 0\hfill & \hfill -{\gamma }^{ij}{n}_{j}{n}_{k}{t}^{k}+{\gamma }^{jk}{n}_{j}{n}_{k}{t}^{i}\hfill \end{array}\right]}^{T}& \text{(30)}\text{}\text{}\\ {\lambda }_{±}={\beta }^{i}{n}_{i}±\alpha \sqrt{{\gamma }^{ij}{n}_{i}{n}_{j}}& ,& {w}_{±}={\left[\begin{array}{cc}\hfill ±\sqrt{{\gamma }^{ij}{n}_{i}{n}_{j}}\hfill & \hfill {\gamma }^{ij}{n}_{j}\hfill \end{array}\right]}^{T}& \text{(31)}\text{}\text{}\end{array}$

#### 3.3 $dk$

(This section very probably contains errors, say Erik on 2005-04-13.)

Setting $\rho ={\mathsc{ℒ}}_{k}u$ with a “Killing” vector ${k}^{a}={\partial }_{t}{x}^{a}$, leading to $\rho =\left(1∕\alpha \right){\partial }_{t}u$.

RHS:

$\begin{array}{rcll}{H}^{ij}& =& {\gamma }^{ij}-\frac{{\beta }^{i}{\beta }^{j}}{{\alpha }^{2}}& \text{(32)}\text{}\text{}\\ {\partial }_{t}u& =& \alpha \rho & \text{(33)}\text{}\text{}\\ {\partial }_{t}\rho & =& \frac{{\beta }^{i}}{\alpha }{\partial }_{i}\alpha \rho +\frac{1}{𝜖}{\partial }_{i}𝜖\left({\beta }^{i}\rho +\alpha {H}^{ij}{v}_{j}\right)& \text{(34)}\text{}\text{}\\ {\partial }_{t}{v}_{i}& =& {\partial }_{i}\alpha \rho & \text{(35)}\text{}\text{}\end{array}$

Propagation matrix:

$\begin{array}{rcll}{A}^{x}& =& \left(\begin{array}{cccc}\hfill 2{\beta }^{x}\hfill & \hfill \alpha \left(-\frac{{\beta }^{x}{\beta }^{x}}{{\alpha }^{2}}+{\gamma }^{xx}\right)\hfill & \hfill \alpha \left(-\frac{{\beta }^{x}{\beta }^{y}}{{\alpha }^{2}}+{\gamma }^{xy}\right)\hfill & \hfill \alpha \left(-\frac{{\beta }^{x}{\beta }^{z}}{{\alpha }^{2}}+{\gamma }^{xz}\right)\hfill \\ \hfill \alpha \hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right)& \text{(36)}\text{}\text{}\end{array}$

Eigensystem: