Discrete Differential Forms for Numerical Relativity

Discrete Differential Forms for Numerical Relativity -...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Discrete Differential Forms for Numerical Relativity Ari Stern Applied & Computational Mathematics California Institute of Technology astern@acm.caltech.edu 24th PCGM March 22, 2008 Stern, Discrete Differential Forms for Numerical Relativity 24th PCGM, 3/22/08 Discrete Differential Forms To specify an n-dimensional mesh, we are given the following topological data for all k = 0 , 1 , . . . , n : • A list of oriented k-cells (vertices, edges, faces, ...), • An incidence matrix ∂ k attaching k-cells to ( k- 1)-cells, with sparse entries ± 1 indicating orientation. A k-chain is a formal linear combination of k-cells. The incidence matrices map k-chains to ( k- 1)-chains, and satisfy ∂ k- 1 ∂ k = 0 . This defines a so-called chain complex . 0-form 1-form 2-form A discrete k-form assigns a degree of freedom to each oriented k-cell of a mesh. This defines a linear map on k-chains; the pairing between forms and chains is denoted by h· , ·i . The discrete exterior derivative is the coboundary operator d k := ∂ * k +1 . Therefore, h dα, σ i = h α, ∂σ i (Stokes’ theorem) holds automatically, as well as d k +1 d k = 0 . 1 Stern, Discrete Differential Forms for Numerical Relativity 24th PCGM, 3/22/08 Successful Application to Computational E&M Let the electromagnetic potential A be a discrete 1-form. Then F = dA is stored on faces, while * F is stored on dual faces.faces....
View Full Document

This note was uploaded on 02/07/2011 for the course PHYS 101 taught by Professor Aster during the Spring '11 term at East Tennessee State University.

Page1 / 7

Discrete Differential Forms for Numerical Relativity -...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online