Computational Relativity

This course develops computational evidence for relativistic problems. It moves from reproducible problem statements through symbolic and spectral calculations to constrained PDE evolution, mesh and performance decisions, horizon and wave extraction, verification, validation, and uncertainty. A plot or converged-looking run is never the endpoint: every claim must remain inside the error that has actually been measured.

The framework supports independent study or local teaching and implies no institution, instructor, schedule, enrollment, computing allocation, or credential.

Required background. The PDE-and-Green-function capability is used to classify principal symbols and characteristics, pose initial and boundary data, select Green operators, and reason about stability. Use PDEs and Green Functions for Gravitation for the scientific core, diagnosis, and review.

Helpful background. A working formulation of Einstein evolution is used to track constraints, gauge, and hyperbolicity; probability and uncertainty reasoning are used to interpret numerical error and model comparison. Review Formulations of General Relativity and Probability and Inference for Gravity for those scientific cores. Use Advanced Classical General Relativity when the formulation background needs a structured review.

Computational-relativity capabilities

By the end, you should be able to:

1. implement or reproduce a relativistic PDE workflow with environment, conventions, inputs, and outputs declared;
2. construct initial data and evolution with constraints, gauge, and boundary conditions explicit;
3. demonstrate convergence, stability, benchmark agreement, and extraction checks; and
4. deliver a clean-rerun capstone whose scientific claims remain within the measured numerical error.

Workload and pacing

Complete the six modules in dependency order, but let verified reruns rather than code size set the pace. Modules 1–4 each carry a numbered assignment; Module 5 carries an extraction-and-error checkpoint; Module 6 consolidates the verification, validation, and uncertainty argument needed downstream. Do not scale a computation before the smaller benchmark and its correction record pass. Run the cross-method transfer task and the capstone as separate clean reruns, including environment, inputs, outputs, and rejection criteria.

Check your preparation

Before using a large code, try a small problem from memory. Use an equivalent accessible response mode or a declared reference aid when needed; the check concerns reconstructible numerical reasoning, not handwriting or speed.

1. Classify the principal part of the wave and Poisson equations and explain the different roles of initial and boundary data.
2. For an explicit finite-difference wave scheme, derive or state its Courant-type stability restriction and distinguish stability from convergence.
3. If $e_h\simeq Ch^p$, derive an observed-order test from three resolutions $h,h/2,h/4$.
4. Explain why satisfying Einstein’s evolution equations approximately does not guarantee that the constraint equations remain small.

Response criteria

1. A secure response identifies the wave equation as hyperbolic and the Poisson equation as elliptic through their principal symbols, assigning initial data to evolution and compatible boundary data to the elliptic problem.
2. A secure response derives the Courant restriction for a stated stencil and integrator, then distinguishes stability from consistency and convergence; it does not quote a universal Courant number without specifying the scheme.
3. A secure response shows that, for $e_h\simeq Ch^p$, the three-level difference ratio approaches $2^p$ in the asymptotic regime and identifies pre-asymptotic and roundoff failure modes.
4. A secure response explains that the constraints restrict admissible data independently of the evolution equations and can be excited by truncation, boundary, gauge, or solver error unless the formulation controls their propagation.

Mark each item secure, uncertain, or not yet. Any not-yet result calls for its preparation review before the dependent module; with no not-yet result, review only the uncertain item; all secure means begin Module 1. Items 1–3 route to the PDE preparation module, which is also the first course step. Item 4 routes to the formulation portions of the advanced-classical path alongside Modules 3 and 4. Direct entry remains available, but a larger simulation does not resolve uncertainty about a numerical principle.

Module 1 — PDE control and reproducible questions

Study in order:

1. PDEs and Green Functions for Gravitation
2. Computational Experiments and Reproducible Workflows

Turn a scientific question into a numerical contract before implementing it. Record equations, formulation, variables, gauge, domain, initial and boundary data, units, conventions, target observable, expected order, environment, precision, tolerances, benchmark, output schema, and failure criteria. Pin machine-readable inputs; do not rely on defaults that can drift.

Assignment 1. Design a reproducibility record for a one-dimensional wave equation with an exact solution. Include a method-of-manufactured-solutions variant, three resolutions, a norm, an expected convergence order, a stability limit, and an automated rejection rule.

Conceptual hint

Organize the record as a continuum problem, a discrete experiment, and predeclared acceptance or rejection criteria.

Setup hint

Specify the exact solution, source, domain, grid, operator, integrator, timestep, norm, sampling times, and three nested resolutions.

Next-step hint

Derive the manufactured source by substitution and use a three-level error ratio tending to $2^p$. Add stability, boundary-contamination, and nonfinite output failure rules.

Model solution architecture

Specify the continuum PDE and source, domain, initial and boundary data, exact solution, grid definition, difference operators, integrator, timestep rule, norm, and sampling times. Insert the exact solution into the PDE to derive the manufactured source rather than guessing it. For $e_h=Ch^p+O(h^{p+1})$, the three-level difference ratio

$$Q=\frac{\lVert u_h-u_{h/2}\rVert} {\lVert u_{h/2}-u_{h/4}\rVert}$$

approaches $2^p$ in the asymptotic regime. Reject the run if the stability limit is violated, nonfinite values occur, boundary reflections enter the measurement window unexpectedly, or the observed order misses a predeclared tolerance. Retain environment and exact command information so a clean rerun can reproduce the table.

Module 2 — Symbolic, ODE, boundary-value, and spectral checks

Study:

3. Symbolic Tensor and Algebraic Computation
4. ODE, Boundary-Value, and Spectral Computation

Symbolic output is evidence only after convention translation, tensor symmetry checks, dimensional or weight checks, and evaluation on an independent metric or limit. Spectral convergence is evidence only after the basis, mapping, regularity, boundary conditions, and conditioning have been examined.

Assignment 2. Compute a curvature scalar for one metric in two coordinate charts, once symbolically and once from an independently coded formula or trusted benchmark. Then solve a regular boundary-value problem by both shooting and a spectral collocation method. Compare invariant output, residuals, conditioning, and convergence.

Conceptual hint

Agreement should be tested through invariants and residuals, not visual similarity of coordinate expressions or solution curves.

Setup hint

Match physical events between charts, keep the curvature implementations independent, and define continuum and boundary residuals for both numerical methods.

Next-step hint

Vary resolution and precision. Report shooting sensitivity separately from collocation conditioning, and verify that the spectral coefficient tail agrees with residual convergence.

Solution criteria

The coordinate expressions may differ, but the scalar must agree at matched events in the overlap domain. The two implementations must not share the same unverified simplification path. For the boundary-value problem, compare the continuum residual and boundary residual, not only the plotted solution. Shooting sensitivity should be reported separately from collocation-matrix conditioning. A coefficient tail can support spectral convergence, but it does not replace comparison under increased resolution and precision.

Module 3 — Relativistic PDEs and initial data

Continue through:

5. Discretizing Relativistic PDEs
6. Numerical Initial Data and Equilibria

Trace the continuum characteristic structure into the discrete boundary treatment. Identify physical, gauge, and constraint fields; distinguish an elliptic constraint solve from hyperbolic evolution; and measure the residual of the actual discrete equations. Initializing auxiliary derivatives from a continuum expression can leave a discrete constraint residual, which must be reported rather than hidden in later error.

Assignment 3. Construct initial data for a constrained toy system by two routes: sampling continuum data and solving the discrete constraint. Evolve both with the same scheme. Compare initial residual, constraint growth, solution error, and convergence, and explain any early-time transient.

Conceptual hint

A continuum solution sampled on a grid need not lie on the discrete constraint surface. Keep that mismatch distinct from later evolution error.

Setup hint

Use the same grid, evolution scheme, norms, and boundaries for sampled data and data solved to a stated discrete tolerance.

Next-step hint

Compare the residual already present at $t=0$, its transient during evolution, and its resolution scaling before deciding whether the remaining difference is phase, amplitude, or constraint error.

Model solution

The sampled continuum solution satisfies the continuum constraint but need not lie on the discrete constraint surface. Its residual should scale with the discretization order if sampling and operators are consistent. The solved data should reduce the discrete residual to the solver tolerance, though its continuum solution error can remain comparable. Evolution may convert the initial residual into a propagating or damped transient. The comparison must use identical norms and boundaries and must not describe the difference as pure phase error unless the constraint component has been separated.

Module 4 — Evolution, gauge, and a benchmark spacetime

Study:

7. Spacetime Evolution, Gauge, and Boundaries

Use the gauge-wave benchmark within the evolution module as an intentionally deceptive test: the continuum spacetime is flat while nontrivial metric components evolve. It tests formulation, gauge, phase, constraint propagation, boundary treatment, and long-time error without physical gravitational radiation.

Assignment 4. Run the benchmark at at least three resolutions for several crossing times. Measure phase error, a metric-field norm, a curvature invariant, constraints, and boundary contamination. Repeat with one admissible change of gauge or formulation parameter and predeclare instability and accuracy failure thresholds.

Conceptual hint

The exact spacetime is flat although the metric components evolve. Separate gauge-dependent field error from phase, constraint, curvature, and boundary diagnostics.

Setup hint

Choose three nested resolutions, several crossing times, one invariant, one constraint norm, a boundary-contamination measure, and fixed failure thresholds before running.

Next-step hint

Compute observed convergence and the first threshold-crossing time, then repeat the same analysis after changing the declared gauge or formulation parameter.

Solution criteria

A successful run shows the expected convergence before boundary or roundoff effects dominate, keeps curvature consistent with flatness within measured error, and separates phase error from amplitude and constraint error. A constraint norm that converges initially but grows exponentially at fixed resolution signals a long-time stability problem. The changed parameter is a sensitivity test, not permission to tune away a failure after inspecting the answer. Report the first time each predeclared threshold is crossed.

Module 5 — Multiscale computing and physical extraction

Continue through:

8. Meshes, Adaptivity, and High-Performance Computing
9. Horizons, Charges, and Gravitational-Wave Extraction

An adaptive or parallel calculation needs both performance and scientific evidence. Show that refinement criteria track the error-relevant structures, that regridding and interprocess boundaries preserve convergence, and that reported speedup does not change the mathematical problem.

For extraction, distinguish coordinate surfaces, quasi-local horizons, finite-radius radiation, extrapolation, and asymptotic constructions. Vary resolution and extraction radius independently so their errors are not conflated.

Checkpoint. Design a two-dimensional error study with numerical resolution on one axis and extraction radius or domain size on the other. State how you would detect an apparently converged waveform whose dominant error is finite-radius extraction.

Module 6 — Verification, validation, and uncertainty

Finish with:

10. Verification, Validation, Uncertainty, and Computational Evidence

Verification asks whether the equations were solved correctly; validation asks whether those equations and inputs represent the intended physical question. Code comparison is useful but not independent evidence when both codes share a formulation, boundary approximation, or benchmark mistake. Classify errors as discretization, iteration, roundoff, boundary, gauge, extraction, model, parameter, or implementation uncertainty and show how each was measured or bounded.

Cross-method transfer task

Take the workflow from the gauge-wave benchmark to one problem with genuine curvature: a scalar field on a fixed black-hole background, a stellar equilibrium perturbation, or a simple constrained spacetime evolution. Before running, revise:

1. the continuum invariant and physical observable;
2. the constraint and boundary diagnostics;
3. the expected convergence regime;
4. the analytic or independent benchmark; and
5. the validation claim.

Explain which checks transfer unchanged and which were specific to the pure gauge solution.

Capstone — clean-rerun relativistic computation

Deliver a bounded calculation that a second reader can rerun in a clean environment. The submission contains:

• a precise scientific question and a claim no broader than the measured observable;
• equations, formulation, gauge, domain, data, units, and conventions;
• locked dependencies, executable command, machine-readable inputs, and checksummed outputs;
• analytic, manufactured, or independently implemented benchmark evidence;
• at least three-resolution convergence and constraint results;
• boundary, extraction, parameter, and precision sensitivity tests;
• a complete uncertainty budget and predeclared failure criteria; and
• a short interpretation of what was verified, what was validated, and what remains model-dependent.

Assess reproducibility, continuum fidelity, numerical stability, convergence, constraint control, independent benchmarking, uncertainty, and claim scope separately. A clean rerun that exposes a failed tolerance is more valuable than an irreproducible “successful” plot.

Continue

Use the Computational Relativity Self-Study Path for a flexible or application-led route. Pair this course with Gravitational Radiation and Compact Binaries for waveforms, Advanced General Relativity and Black-Hole Physics for formulations, or a Research record when the objective is to reproduce and reassess a current claim.