Why numerical relativity? The field equations Code ecosystem and numerical methods The Methods of Numerical Relativity David Hilditch Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena Octopus Developers Meeting January 29, 2015 David Hilditch The Methods of Numerical Relativity Summary Why numerical relativity? The field equations Code ecosystem and numerical methods Outline Why numerical relativity? The field equations Code ecosystem and numerical methods Summary David Hilditch The Methods of Numerical Relativity Summary Why numerical relativity? The field equations Code ecosystem and numerical methods When not to try Numerical Relativity We seek solutions to the gravitational field equations, Gab = Rab − 12 gab R = 8πTab . which describe how curvature of spacetime evolves and is related to matter distribution. Non-linear PDEs. How can we find solutions? Different physics: • Is the desired solution time independent, or with a lot of symmetry? There might be an exact solution. Read a book. • Is the desired solution close to an exact solution that we already know? Perturbation theory/asymptotic expansions. Second-order? David Hilditch The Methods of Numerical Relativity Summary Why numerical relativity? The field equations Code ecosystem and numerical methods When to try Numerical Relativity On the other hand: • Are the fields highly dynamical? Are non-linear terms expected to dominate the behavior? Is there no symmetry? Use Numerical Relativity; numerical methods to find approximate solutions that converge to the continuum solution in the limit of infinite resolution. David Hilditch The Methods of Numerical Relativity Summary Why numerical relativity? The field equations Code ecosystem and numerical methods Summary Ingredients Mathematics: • PDEs theory. • Differential geometry. • Numerical analysis. Physics: • GR. • Matter models. Computation: • HPC. • Numerical methods. Event horizon of merging blackholes. Thierfelder. 2009 • Visualization. David Hilditch The Methods of Numerical Relativity Why numerical relativity? The field equations Code ecosystem and numerical methods Summary Example problems • Critical collapse. Can we make arbitrarily small blackholes? • Computational astrophysics: supernova core collapse. • Gravitational wave astronomy. Curvature in GW collapse evolution. DH, Weyhausen, Brügmann. 2015 David Hilditch The Methods of Numerical Relativity Why numerical relativity? The field equations Code ecosystem and numerical methods Example problems • Critical collapse. Can we make arbitrarily small blackholes? • Computational astrophysics: supernova core collapse. • Gravitational wave astronomy. LIGO Hanford David Hilditch The Methods of Numerical Relativity Summary Why numerical relativity? The field equations Code ecosystem and numerical methods Summary Example problems t/M = 428.264 −3 8 • Critical collapse. Can we −4 make arbitrarily small blackholes? supernova core collapse. 4 −4.5 2 y/M • Computational astrophysics: −3.5 Z4c 6 −5 0 −5.5 −6 −2 −6.5 −4 −7 • Gravitational wave −6 −7.5 −8 −8 astronomy. −6 −4 −2 0 x/M 2 4 6 8 Contour plot of density of NS. DH+. 2013 David Hilditch The Methods of Numerical Relativity −8 Why numerical relativity? The field equations Code ecosystem and numerical methods Summary A brief history of NR From Wikipedia NR page and Smarr: • Late 1950s: ADM decomposition. • 1960s. First numerics, including Smarr [nice talk: http://online.itp.ucsb.edu/online/numrel00/smarr/]. • 1970s. Axisymmetric spacetimes. York 3+1 decomposition. • 1980s. Numerical Relativity becomes a field. • 1990s. Critical Phenomena. 3d work becomes feasible. Grand challenge. • 2000s. Compact binaries. David Hilditch The Methods of Numerical Relativity Why numerical relativity? The field equations Code ecosystem and numerical methods Outline Why numerical relativity? The field equations Code ecosystem and numerical methods Summary David Hilditch The Methods of Numerical Relativity Summary Why numerical relativity? The field equations Code ecosystem and numerical methods Summary Evolution equations and constraints Decomposition: ∂t γij = −2αKij + 2D(i βj) , k ∂t Kij = −Di Dj α + α[Rij − 2K i Kjk + KKij ] + Lβ Kij , H = R − Kij K ij 2 + K = 0, ρ=4/ρ=8 ρ=8/ρ=16 ρ=16/ρ=32 ρ=32/ρ=64 ρ=64/ρ=128 ρ=128/ρ=256 6 ρ=256/ρ=512 ρ=512/ρ=1024 10 8 j Mi = D (Kij − γij K ) = 0 . 4 • Evolution and constraint equations. 2 0 • Constraint subsystem closed. -2 -4 • Gauge freedom similar to E&M / YM. 0 0.1 0.2 0.3 0.4 0.5 Cao and DH. 2011 Formulation crucial! David Hilditch The Methods of Numerical Relativity 0.6 0.7 0.8 Why numerical relativity? The field equations Code ecosystem and numerical methods Summary Evolution equations and constraints Decomposition: ∂t γij = −2αKij + 2D(i βj) , k ∂t Kij = −Di Dj α + α[Rij − 2K i Kjk + KKij ] + Lβ Kij , H = R − Kij K ij 2 + K = 0, ρ=4/ρ=8 ρ=8/ρ=16 ρ=16/ρ=32 ρ=32/ρ=64 ρ=64/ρ=128 ρ=128/ρ=256 6 ρ=256/ρ=512 ρ=512/ρ=1024 10 8 j Mi = D (Kij − γij K ) = 0 . 4 • Evolution and constraint equations. 2 0 • Constraint subsystem closed. -2 -4 • Gauge freedom similar to E&M / YM. 0 0.1 0.2 0.3 0.4 0.5 Cao and DH. 2011 Formulation crucial! David Hilditch The Methods of Numerical Relativity 0.6 0.7 0.8 Why numerical relativity? The field equations Code ecosystem and numerical methods Outline Why numerical relativity? The field equations Code ecosystem and numerical methods Summary David Hilditch The Methods of Numerical Relativity Summary Why numerical relativity? The field equations Code ecosystem and numerical methods Summary Methods Bare-bones summary: Grids: • Grid-up (physical). • Time integration: MoL. • Moving-box/adaptive MR. • Multipatch. Spatial derivatives: • Corotation. • Finite differences. Popular codes: • HRSC. (Fluids). • The Einstein Toolkit • Pseudospectral. • Spectral Einstein Code. Boundaries: • BAM and bamps. (Jena) • Elliptic for ID. Various licences. • CPBCs. David Hilditch The Methods of Numerical Relativity Why numerical relativity? The field equations Code ecosystem and numerical methods Summary Parallelization Implementation typically with: • Automated generation. Parallelization: • MPI/OMP standard. 102 Time units per hour • Mixture of C++ /C/Fortran. Ideal Scaling SuperMUC - only MPI SuperMUC - 2 OMP/MPI SuperMUC - 4 OMP/MPI quadler - only MPI quadler - 2 OMP/MPI quadler - 4 OMP/MPI 101 • GPU? Memory a problem. Some applications. 100 • MR & balancing an issue. 96 192 384 768 1536 Number of cores Strong scaling of BAM at LRZ. David Hilditch The Methods of Numerical Relativity 3072 6144 Why numerical relativity? The field equations Code ecosystem and numerical methods Summary Parallelization Implementation typically with: 10 • Mixture of C++ /C/Fortran. Parallelization: • MPI/OMP standard. • GPU? Memory a problem. Some applications. • MR & balancing an issue. David Hilditch Time units per hour • Automated generation. 1.0 Ideal Scaling SuperMUC 0.1 279 558 1115 Number of cores 2230 Strong scaling of bamps at LRZ. The Methods of Numerical Relativity 4459 Why numerical relativity? The field equations Code ecosystem and numerical methods Outline Why numerical relativity? The field equations Code ecosystem and numerical methods Summary David Hilditch The Methods of Numerical Relativity Summary Why numerical relativity? The field equations Code ecosystem and numerical methods Summary Some parting thoughts... Points for discussion: • Solid mathematical foundation, although implementation of principle solutions not always clear. • Smooth solutions often expected. Meaningful error analysis possible! • Perfect parallelization hard. Libraries? One expects a great deal of overlap with the rest of computational physics. Sadly much develoment in isolation. David Hilditch The Methods of Numerical Relativity
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