Project: Hyperbolic Relativity
Project: Hyperbolic Relativity We have developed a new formulation of the Einstein equations that casts them in an explicitly first order, flux-conservative, hyperbolic form:
Here u is a vector of quantities related with the gravitational field and F and S are the flux and source vector terms, which do not contain any derivative. The system is strictly hyperbolic as the jacobian of F can be diagonalized and we can find the complete set of eigenvectors and eigenfields of the system. Based on previous work we show that this now can be done for a wide class of time slicing conditions, including maximal slicing, making it very useful for numerical relativity. This development permits the application to the Einstein equations of advanced numerical methods developed to solve the fluid dynamic equations, without overly restricting the time slicing, for the first time. The theoretical formulation is given in: A New Formalism for Numerical Relativity .
Check the slides of the talk "Hyperbolic Relatitivy" presented by Joan Massó at the Third Texas Workshop on 3-dimensional Numerical Relativity
Results with a new 1D code ( Nage , developed by C. Bona, J. Massó and J. Stela) that evolves a spherically symmetric black hole show how we can tremendously improve the results over the standard approach. Numerical errors make the mass of the black hole to from its theoretical value and eventually lead a code to crash. A code using the standard ADM approach crashes at times of around 100M (where M is the mass of the black hole), with an error of more than 100% on the mass. The new formulation allows evolution past 1000 M with errors around 2% at that late time. Large gradients developing in the metric functions are treated as a shock, using flux limiters and upwinding in the direction of propagation.
Recently, we have added support of a shift vector using the idea of transforming the system with a shift to a "tilde" system without a shift.
We have also developed a full 3D code that applies all this new stuff to black holes and gravitational waves. The are a lot of exciting news but now I'm too lazy to explain them... check soon.. Check also the reports in the H and NewAge project page.
Several publications have been completed on this work, and several more are in progress. Check this cronological list of Related papers on our server:
- C. Bona, J. Massó,
Hyperbolic system for Numerical Relativity,
Phys. Rev. Lett. 68, 1097 (1992).- C. Bona, J. Massó,
A vacuum, fully relativistic 3D numerical code,
in ``Approaches to Numerical Relativity'', Ray d'Inverno (ed.), Cambridge University Press, 1992.- C. Bona, J. Massó,
Numerical Relativity: evolving space-time,
International Journal of Modern Physics C: Physics and Computers 4, 883 (1993)- C. Bona, J.M. Ibáñez, J.M. Martí and J. Massó, Shock capturing methods in 1-D Numerical Relativity,
in ``Gravitation and General Relativity: rotating bodies and other topics'', F. Chinea (ed.), Lecture Notes in Physics, vol. 423, Springer Verlag, 1993.- C. Bona, J. Massó, J. Stela,
Numerical Black Holes: a Moving Grid Approach,
Phys. Rev. D., 51, 1639 (1995)
gr-qc/9412070- C. Bona, J. Stela, J. Massó, E. Seidel,
Invariant Algebraic Slicing of the Spacetime ,
to appear in the proceedings of the ERE 94, Menorca, Spain (World Scientific, 1995).
gr-qc/9412061- C. Bona, J. Massó, E. Seidel, J. Stela,
A New Formalism for Numerical Relativity ,
Phys. Rev. Letters 75, 600 (1995)
gr-qc/9412071
For a description of the codes under development, check the project H and NewAge .See also the
H3expresso page for a special version of the H code.