Numerical Boundary Conditions

A numerical outer boundary is an active part of a relativistic evolution, not a distant line where values may be chosen freely. Hyperbolic characteristic fields carry information either into or out of the domain. Prescribing an outgoing field overconstrains the problem; failing to control an incoming constraint field can destabilize it. A boundary implementation must therefore be derived from characteristics and tested by reflection, energy, and constraint diagnostics.

Required background. Ordinary Differential Equations supplies semidiscrete systems and stability of boundary-coupled evolution.

Helpful background. Formulations of General Relativity supplies characteristic and constraint fields, while Partial Differential Equations and Green Functions provides well-posed boundary data and outgoing-wave behavior.

Prescribe only incoming fields

For the first-order wave system

$$\partial_t\Pi=\partial_x\Phi, \qquad \partial_t\Phi=\partial_x\Pi,$$

the combinations $w_\pm=\Pi\pm\Phi$ obey

$$\partial_tw_+-\partial_xw_+=0, \qquad \partial_tw_-+\partial_xw_-=0.$$

Thus $w_+$ travels left at speed $-1$ and $w_-$ travels right at speed $+1$. At the right boundary $w_+$ is incoming; at the left boundary $w_-$ is incoming. A maximally dissipative condition has the schematic form

$$w_{\mathrm{in}}=q\,w_{\mathrm{out}}+g, \qquad \lvert q\rvert\le1,$$

where $g$ is supplied data. Setting the incoming radiative field to zero absorbs the normal planar wave in this system. The analogous local condition is generally only approximate for multipoles on a finite spherical boundary.

For this constant-coefficient one-dimensional system it is exactly absorbing. Indeed, $E=\tfrac12\int(\Pi^2+\Phi^2)dx$ satisfies $\dot E=[\Pi\Phi]_{\rm left}^{\rm right}$ and $\Pi\Phi=(w_+^2-w_-^2)/4$. At the right boundary, homogeneous data $w_+=q w_-$ contribute $(q^2-1)w_-^2/4\le0$ for real $\lvert q\rvert\le1$. The same outward-flux argument applies at the left boundary with incoming and outgoing interchanged.

The numerical closure—ghost cells, penalty terms, or interface fluxes—must reproduce the continuum energy sign. Count incoming fields before specifying data. Applying both Dirichlet and Neumann conditions to the same incoming characteristic generally overprescribes the system.

Constraint-preserving and radiative data

Formulations of Einstein’s equations include physical, gauge, and constraint characteristic sectors. Boundary conditions may control incoming constraints while specifying gauge and reducing incoming gravitational radiation. These roles must be kept distinct; setting every incoming evolved variable to zero need not preserve the constraints.

For a generalized-harmonic system, the decomposition into constraint, physical, and gauge boundary conditions and the corresponding projected equations are given in Rinne 2006, § 2.1, Eqs. 9–11; § 2.2, Eqs. 26 and 28; and § 2.3, Eqs. 30–34. Translate normal orientation, curvature, and null-tetrad signs before implementing those formulas. The continuum derivation determines which characteristic field is incoming; the discretization determines how it is imposed stably.

At an excision boundary inside a black-hole horizon, all physical characteristics may leave the computational domain for a suitable gauge, so no physical data are prescribed. Verify the actual coordinate characteristic speeds during evolution rather than assuming causal intuition guarantees pure outflow.

Outgoing-pulse benchmark

For an explicit one-dimensional closure test, use domains $0\leq x\leq R$ with $R=6,8,10$ and the smooth compact bump

$$B(x)= \begin{cases} \exp\!\left[1-\dfrac{1}{1-s^2}\right],&\lvert s\rvert\lt 1,\\ 0,&\lvert s\rvert\geq1, \end{cases} \qquad s=\frac{x-2}{0.5}.$$

Initialize the right-moving pulse $\Pi=-B'$ and $\Phi=B'$. Evolve the characteristic equations with the fourth-order-interior, second-order-boundary diagonal-norm SBP derivative, classical RK4, and $\Delta t/h=0.2$. At the left boundary impose $w_-=0$. At the right, run both $w_+=0$ and $w_+=q w_-$ with the known reflection amplitude $q=1/4$. For endpoint selector vectors $e_0,e_N$, the weak enforcement is

$$\begin{aligned} \dot{\mathbf w}_-&=-D\mathbf w_--H^{-1}e_0(w_-)_0,\\ \dot{\mathbf w}_+&=D\mathbf w_+-H^{-1}e_N[(w_+)_N-q(w_-)_N]. \end{aligned}$$

Use 64, 128, and 256 subintervals per unit length. Differentiate the displayed bump analytically when constructing the initial data; a grid derivative would make initialization error part of the boundary measurement. The $q=0$ case is a useful null test, but in exactly decoupled characteristic equations an initially zero $w_+$ remains zero regardless of outgoing phase error. The nonzero-$q$ case actually exercises the boundary coupling and return path.

At the fixed extraction point $x=4$, measure

$$\mathcal R= \frac{\lVert w_{\mathrm{incoming}}\rVert} {\lVert w_{\mathrm{outgoing}}\rVert},$$

using time $L^2$ norms on the outgoing window $1.35\leq t\leq2.65$ and, for each $R$, the reflected window $2R-6.65\leq t\leq2R-5.35$. Continue each run to $t=2R-5$. The exact characteristic solution, with $B'$ extended by zero outside its support, is

$$w_-(t,x)=-2B'(x-t), \qquad w_+(t,x)=-2qB'(2R-x-t).$$

It predicts $\mathcal R=\lvert q\rvert$. Recover $\Pi=(w_++w_-)/2$ and $\Phi=(w_+-w_-)/2$ for field-error checks. An $R=20$ run at 512 subintervals per unit length provides a second outgoing-pulse reference before its boundary can return a signal; it is not the reference for the deliberately reflected $q=1/4$ pulse. Report field error, boundary residual, discrete energy, and $\mathcal R$ at every $R$ and resolution. The $q=0$ return should vanish, while the $q=1/4$ amplitude and shape should converge to the exact reflection. A spherical local condition has a separate finite-radius modeling error and needs its own wavelength-to-radius sweep.

For a Schwarzschild perturbative or full-relativity test, compare extracted radiation with a larger-domain reference and track the incoming curvature. In an outward-adapted null tetrad that incoming radiative Weyl scalar is $\Psi_0$; $\Psi_4$ represents the opposite propagation direction under the same convention. Rinne, Lindblom, and Scheel 2007, §§ 2–4 and §§ 5.1–5.2 tests constraint-preserving conditions that freeze $\Psi_0$ and quantifies reflections. Translate the source’s tetrad and curvature signs before comparing to code variables. This page specifies the numerical test; it does not replace the continuum radiation interpretation.

Periodic, matching, and interface boundaries

Periodic boundaries are excellent interior benchmarks because they remove physical outer-boundary modeling, but they are inappropriate when outgoing radiation should escape. Matching to a characteristic or exterior domain can reduce finite-radius contamination while adding interpolation and interface conditions. Treat the matching surface like any other two-sided flux: verify conservation or energy transfer and convergence of interpolation residuals.

Moving the outer boundary is an independent diagnostic. If a feature arrives later in proportion to boundary radius, it is likely reflected contamination. If it remains at the same physical time under radius changes, inspect the interior scheme instead. Boundary-radius variation should accompany resolution variation because the two errors scale differently.

Reproducible environment and failure criteria

Pin the implementation revision, formulation and characteristic normalization, outward-normal orientation, boundary radius, closure or penalty coefficients, absorbing order, gauge data, spatial operator, time integrator, precision, and extraction location. A clean rerun exports characteristic speeds, incoming and outgoing fields, energy and constraint histories, reflection coefficient, boundary-radius and resolution sweeps, and the large-domain comparison.

The validity regime includes the declared characteristic-speed signs and wavelength-to-radius range. Failure modes include overprescription, an unstable penalty sign, constraint injection hidden by a small field error, incoming modes left uncontrolled, and a reflection test contaminated by overlapping pulses. Reject a boundary that converges only while it is causally disconnected from the measurement interval. Also reject a “no boundary effect” claim made without moving the boundary or comparing a larger domain.

Check your understanding

At a right boundary, one wave characteristic exits and one enters. A code fixes both to analytic values and obtains a smooth short run. Why is this not yet a valid boundary treatment?

Solution

Only the incoming characteristic may be freely prescribed. Fixing the outgoing field can overconstrain the continuum problem and produce reflection or instability once numerical error reaches the boundary. Reformulate the condition in characteristic variables, impose data only on incoming sectors, and verify energy, reflection, and constraint convergence over a long enough interval.

References

• Rinne, O. (2006), “Stable Radiation-Controlling Boundary Conditions for the Generalized Harmonic Einstein Equations,” Classical and Quantum Gravity 23, 6275–6300. arXiv.
• Rinne, O., Lindblom, L., and Scheel, M. A. (2007), “Testing Outer Boundary Treatments for the Einstein Equations,” Classical and Quantum Gravity 24, 4053–4078. arXiv.