General relativistic null-cone evolutions with a high-order scheme

Abstract

We present a high-order scheme for solving the full non-linear Einstein equations on characteristic null hypersurfaces using the framework established by Bondi and Sachs. This formalism allows asymptotically flat spaces to be represented on a finite, compactified grid, and is thus ideal for far-field studies of gravitational radiation. We have designed an algorithm based on 4th-order radial integration and finite differencing, and a spectral representation of angular components. Consequently the scheme offers more accuracy at a given computational cost compared to previous methods which are second-order accurate. Based on a newly implemented code, we show that the new numerical scheme remains stable and is convergent at the expected order of accuracy.

Appendices

Appendix A: The nonlinear terms in the Einstein equations

The nonlinear terms $N_\beta , N_Q, N_U, N_W$ and $N_J$ in (16a) through (17) were first presented in [12]. We repeat them here, but with a mis-print in Eq. (A3) of [12] corrected.

$$\begin{aligned} N_\beta&= \frac{r}{8}\left( J_{,r}\bar{J}_{,r}-K^2_{,r} \right) .\end{aligned}$$

(61)

$$\begin{aligned} N_U&= \frac{e^{2\beta }}{r^2} \left( KQ-Q-J\bar{Q} \right) ,\end{aligned}$$

(62)

$$\begin{aligned} N_Q&= r^2 \Bigg ( (1-K) ( \eth K_{,r} + {\bar{\eth }} J_{,r} ) + \eth (\bar{J} J_{,r} ) + {\bar{\eth }} ( J K_{,r} ) - J_{,r} {\bar{\eth }} K \nonumber \\&+ \frac{1}{2K^2}(\eth \bar{J} (J_{,r} - J^2 \bar{J}_{,r} ) + \eth J (\bar{J}_{,r} -\bar{J}^2 J_{,r} ) ) \Bigg ).\end{aligned}$$

(63)

$$\begin{aligned} N_W&= e^{2 \beta } \Bigg ( (1-K) ( \eth {\bar{\eth }} \beta + \eth \beta {\bar{\eth }} \beta ) + \frac{1}{2} \bigg ( J ({\bar{\eth }} \beta )^2 + \bar{J} (\eth \beta )^2 \bigg ) \nonumber \\&- \frac{1}{2} \bigg ( \eth \beta ( {\bar{\eth }} K - \eth \bar{J}) + {\bar{\eth }} \beta ( \eth K - {\bar{\eth }} J ) \bigg ) + \frac{1}{2} \bigg ( J {\bar{\eth }}^2 \beta + \bar{J} \eth ^2 \beta \bigg ) \Bigg ) \nonumber \\&- e^{-2 \beta } \frac{r^4}{8} ( 2 K U_{,r} \bar{U}_{,r} + J \bar{U}^2_{,r} + \bar{J} U^2_{,r}).\end{aligned}$$

(64)

$$\begin{aligned} N_J&= N_{J1}\!+\!N_{J2}\!+\!N_{J3}\!+\!N_{J4}\!+\!N_{J5}\!+\!N_{J6}\!+\!N_{J7}\!+\!\frac{J}{r}(P_1\!+\!P_2 \!+\!P_3\!+\!P_4) \nonumber \\ \end{aligned}$$

(65)

where

$$\begin{aligned} N_{J1}&= - \frac{e^{2 \beta }}{r} \bigg ( K ( \eth J {\bar{\eth }} \beta + 2 \eth K \eth \beta - {\bar{\eth }}bar \eth J \eth \beta ) \!+\! J ( {\bar{\eth }}bar \eth J {\bar{\eth }}bar \eth \beta \nonumber \\&\quad - 2 \eth K {\bar{\eth }}bar \eth \beta ) - \bar{J} \eth J \eth \beta \bigg ) , \nonumber \\ N_{J2}&= -\frac{1}{2} \bigg ( \eth J ( r \bar{U}_{,r} \!+\! 2 \bar{U}) \!+\! {\bar{\eth }}bar \eth J ( r U_{,r} + 2 U) \bigg ) , \nonumber \\ N_{J3}&= (1-K) ( r \eth U_{,r} \!+\! 2 \eth U) - J ( r \eth \bar{U}_{,r} \!+\! 2 \eth \bar{U}) , \nonumber \\ N_{J4}&= \frac{r^3}{2} e^{-2 \beta } \bigg ( K^2 U^2_{,r} \!+\! 2 J K U_{,r} \bar{U}_{,r} \!+\! J^2 \bar{U}^2_{,r} \bigg ) , \nonumber \\ N_{J5}&= - \frac{r}{2} J_{,r} ( \eth \bar{U} \!+\! {\bar{\eth }} U) , \nonumber \\ N_{J6}&= r \Bigg ( \frac{1}{2} ( \bar{U} \eth J + U {\bar{\eth }} J ) (J \bar{J}_{,r} - \bar{J} J_{,r} ) \nonumber \\&+ ( J K_{,r} - K J_{,r} ) \bar{U} {\bar{\eth }} J - \bar{U} ( \eth J_{,r} - 2 K \eth K J_{,r} + 2 J \eth K K_{,r} ) \nonumber \\&- U ( {\bar{\eth }}J_{,r} - K \eth \bar{J} J_{,r} \!+\! J \eth \bar{J} K_{,r} ) \Bigg ) , \nonumber \\ N_{J7}&= r ( J_{,r} K - J K_{,r} ) \bigg ( \bar{U} ( {\bar{\eth }} J - \eth K ) + U ( {\bar{\eth }} K - \eth \bar{J} ) \nonumber \\&+ K ( {\bar{\eth }} U - \eth \bar{U} ) \!+\! ( J {\bar{\eth }} \bar{U} - \bar{J} \eth U ) \bigg ) , \nonumber \\ P_1&= r^2 \bigg ( \frac{J_{,u}}{K} (\bar{J}_{,r} K - \bar{J} K_{,r} ) \!+\! \frac{\bar{J}_{,u}}{K} ( J_{,r} K - J K_{,r} ) \bigg ) - 8 \left( r+r^2\hat{W}\right) \beta _{,r} , \nonumber \\ P_2&= e^{2 \beta } \Bigg ( - 2 K ( \eth {\bar{\eth }} \beta + {\bar{\eth }} \beta \eth \beta ) - ( {\bar{\eth }} \beta \eth K + \eth \beta {\bar{\eth }} K) \nonumber \\&+ \bigg ( J ( {\bar{\eth }}^2 \beta + ({\bar{\eth }} \beta )^2 ) + \bar{J} ( \eth ^2 \beta + (\eth \beta )^2 ) \bigg ) + ( {\bar{\eth }} J {\bar{\eth }} \beta + \eth \bar{J} \eth \beta ) \Bigg ), \nonumber \\ P_3&= \frac{r}{2} \bigg ( ( r {\bar{\eth }} U_{,r} + 2 {\bar{\eth }} U) + ( r \eth \bar{U}_{,r} + 2 \eth \bar{U}) \bigg ) , \nonumber \\ P_4&= - \frac{r^4}{4} e^{-2 \beta } ( 2 K U_{,r} \bar{U}_{,r} + J \bar{U}^2_{,r} + \bar{J} U^2_{,r} ). \end{aligned}$$

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Appendix B: Regularized equations at ${\mathcal{J }^{+}}$

At ${\mathcal{J }^{+}}$, the equations simplify when inserting Eqs.  (21) and (22) and taking the limit $r\rightarrow \infty$. In particular, terms containing powers of $1/r,...,1/r^n$ vanish. Below, we give the equations evaluated at ${\mathcal{J }^{+}}$.

$$\begin{aligned} \beta _{,x}&= 0, \end{aligned}$$

(67)

$$\begin{aligned} Q&= -2\eth \beta , \end{aligned}$$

(68)

$$\begin{aligned} U_{,x}&= r_\Gamma ^{-1}e^{2\beta }(KQ-J\bar{Q}), \end{aligned}$$

(69)

$$\begin{aligned} \hat{W}&= \frac{\eth \bar{U}+{\bar{\eth }} U}{2}, \end{aligned}$$

(70)

$$\begin{aligned} \Phi&= -\eth U + N_{J,{\mathcal{J }^{+}}}, \end{aligned}$$

(71)

where

$$\begin{aligned} N_{J,{\mathcal{J }^{+}}} = - \frac{\bar{U} \eth J + U \bar{\eth }J}{2} + (1-K)\eth U - \frac{J(\eth \bar{U}-\bar{\eth }U)}{2}. \end{aligned}$$

(72)

Appendix C: Radial derivative operators

We use fourth-order sidewinded derivatives

$$\begin{aligned} {\partial }f_i&= \frac{1}{\Delta x}\left( -\frac{25}{12}f_i + 4 f_{i+1} - 3 f_{i+2} + \frac{4}{3}f_{i+3} - \frac{1}{4}f_{i+4}\right) , \end{aligned}$$

(73)

$$\begin{aligned} {\partial }^2 f_i&= \frac{1}{\Delta x^2}\left( \frac{15}{4}f_i - \frac{77}{6}f_{i+1} + \frac{107}{6}f_{i+2} - 13 f_{i+3} + \frac{61}{12}f_{i+4} - \frac{5}{6}f_{i+5}\right) ,\nonumber \\ \end{aligned}$$

(74)

where $\Delta x$ is the grid spacing in the compactified radial coordinate direction.

Close to ${\mathcal{J }^{+}}$ when $i > N_x-5$, we switch to 4th-order centred stencils

$$\begin{aligned} {\partial }f_i&= \frac{1}{\Delta x}\left( +\frac{1}{12}f_{i-2} - \frac{2}{3} f_{i-2} +\frac{2}{3}f_{i+1} - \frac{1}{12}f_{i+2}\right) , \end{aligned}$$

(75)

$$\begin{aligned} {\partial }^2 f_i&= \frac{1}{\Delta x^2}\left( -\frac{1}{12}f_{i-2} + \frac{4}{3}f_{i-1} -\frac{5}{2}f_{i} + \frac{4}{3} f_{i+1} - \frac{1}{12}f_{i+2}\right) , \end{aligned}$$

(76)

and when $i > N_x-3$, we switch to side-winded stencils pointing towards the inner boundary

$$\begin{aligned} {\partial }f_i \!&= \! \frac{1}{\Delta x}\left( \frac{25}{12}f_i\! -\! 4 f_{i-1} \!+\! 3 f_{i-2} \!-\! \frac{4}{3}f_{i-3} \!+\! \frac{1}{4}f_{i-4}\right) , \end{aligned}$$

(77)

$$\begin{aligned} {\partial }^2 f_i \!&= \! \frac{1}{\Delta x^2}\left( \frac{15}{4}f_i \!-\! \frac{77}{6}f_{i-1} \!+\! \frac{107}{6}f_{i-2} \!-\! 13 f_{i-3} \!+\! \frac{61}{12}f_{i-4} \!-\! \frac{5}{6}f_{i-5}\right) . \end{aligned}$$

(78)

Appendix D: Dissipation operator stencil

We use a 5th-order Kreiss–Oliger dissipation operator of the form

$$\begin{aligned} D f_i \!=\! \frac{\epsilon _\mathrm{diss}}{64 \Delta x} \left( f_{i-3} \!- \!6 f_{i-2} \!+\! 15 f_{i-1} \!-\! 20 f_{i} \!+\! 15 f_{i+1} \!-\! 6 f_{i+2} \!+\! f_{i+3}\right) \!,\quad \end{aligned}$$

(79)

where $\epsilon _\mathrm{diss}$ controls the strength of the applied dissipation operator $D$.

Close to the outer boundary where we do not have enough points to apply the standard centred-stencil Kreiss–Oliger dissipation operator Eq. (79), we use side-winded dissipation operator stencils derived for SBP operator (though we do not make explicit use of the SBP property) $D_{4-2}$ of [37]. This particular dissipation operator is defined via coefficients $a_{ij}$ and $q_{i}$. In the interior of the domain the operator reads

$$\begin{aligned} A_{ij} u_j&= \frac{\epsilon _\mathrm{diss}}{2^{2p}} \left[ q_0 u_i + \sum _{j=1}^7 q_{j} \left( u_{i-j} + u_{i+j} \right) \right] \end{aligned}$$

(80)

where the $q_i$ are the coefficients given in Eq. (79). Near the outer boundary (on points $i=N_x-5,...,N_x-1$), the operator can be written as

$$\begin{aligned} A_{ij} u_j&= \frac{\epsilon _\mathrm{diss}}{2^{2p}} \sum _{j=1}^{7} a_{j,N_x-i} u_{N_x-j}, \end{aligned}$$

(81)

where the coefficients $a_{ij}$ are taken from [37] reported below. We note that $\epsilon _\mathrm{diss}\ge 0$ selects the amount of dissipation and is usually of order unity.

$$\begin{aligned} (a_{ij}) = \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} -\frac{48}{17} &{} \frac{144}{17} &{} -\frac{144}{17} &{} \frac{48}{17} &{} 0 &{} 0 &{} 0 \\ \frac{144}{59} &{} -\frac{480}{59} &{} \frac{576}{59} &{} -\frac{288}{59} &{} \frac{48}{59} &{} 0 &{} 0 \\ \frac{144}{43} &{} \frac{576}{43} &{} -\frac{912}{43} &{} \frac{720}{43} &{} -\frac{288}{43} &{} \frac{48}{43} &{} 0 \\ \frac{48}{49} &{} -\frac{288}{49} &{} \frac{720}{49} &{} -\frac{960}{49} &{} \frac{720}{49} &{} -\frac{288}{49} &{} \frac{48}{49} \end{array}\right) \end{aligned}$$

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