Fundamental Issues of Numerical Relativity

Current  UBC Group

Outline

Something to Contemplate …

Relativists’ Units

Target Physics

Gravitational Radiation

 Target Physics

Assumptions / Restrictions
(for this talk)

Nature of Relativistic Gravitation

Black Holes and Physical Singularities

Singularity Avoidance: Black Hole Excision

Core Mathematical Problem
“They’re just a set of PDEs, Larry”

The (Fundamentally) Good News

The (Fundamentally) Good News

The (Fundamentally) Good News

A Principle One Can Take to the Bank
(Brandt’s “Golden Rule”)

Optimization / Efficiency

Flip Side of The Golden Rule
(The No Free Lunch Principle)

The “3+1” split

3+1 EOM (schematic)

3+1 EOM (schematic)

Update Possibilities

Update Possibilities

Coordinate Pathologies

Building “Good” Coordinate Systems

Boundary Conditions (Outer)

Boundary Conditions (Outer)

Boundary Conditions (Inner)

Discretization Issues

Computational Demands

Computational Demands

Computational Demands

Model Problems

Model Problems

Convergence and Stability

Convergence

Stability

Stability and Dissipation

Kreiss-Oliger Dissipation: Example

Kreiss-Oliger Dissipation: Example

Effect of Dissipation
65 x 129 Grid
(Axisymmetric collapse of scalar field)

Effect of Dissipation
65 x 129 Grid

Effect of Dissipation
65 x 129 Grid

Effect of Dissipation
129 x 257 grid

Effect of Dissipation
129 x 257 grid

Effect of Dissipation
129 x 257 grid

Effect of Dissipation
127 x 259 grid

Resolution

Adaptive Mesh Refinement
(Pretorius, Choptuik, Hirschmann and Liebling;
following Berger & Oliger, JCP, 1984)

Adaptive Mesh Refinement
(near-critical axisymmetric collapse of scalar field)

Adaptive Mesh Refinement
(near-critical axisymmetric collapse of scalar field)

Validation / Error Analysis

Validation / Error Analysis

Validation / Error Analysis

Validation / Error Analysis

Other Issues

Other Issues