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Peano
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Peano
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ExaGRyPE is a numerical relativity application built based on the engine ExaHyPE2, whose current version conduct simulation of vacuum and black hole spacetimes. ExaGRyPE solves the Einstein field equations in the standard CCZ4 formulation under a 3+1 foliation, and employs a block-structured Cartesian grid carrying a higher-order Finite Difference scheme with full support of adaptive mesh refinement (AMR). It facilitates massive parallelism combining message passing, domain decomposition and task parallelism, and it supports the injection of particles into the grid as static data probes or as moving tracers.
ExaGRyPE is actively under development as a part of the ExaHyPE projects and is evolving as a novel numerical relativity code within the computational astrophysics community. Planned enhancements include optimizing its performance, porting it to GPUs, and extending its capabilities to simulate more complex scenarios beyond black hole spacetimes. These advancements will involve incorporating matter components via standard Euler fields or Smoothed-Particle Hydrodynamics (SPH). Additionally, thanks to the high flexibility in dimensionality offered by ExaHyPE2, future versions will enable full simulations of modified gravity theories in higher dimensions.
The release paper of ExaGRyPE is provided below for interested users. The code is completely open-sources and can be found in the Peano repositories. How to Use ExaGRyPE provide a tutorial on how to install and run the code.
The ExaGRyPE code solves the Einstein equations in the Z4 formulation
\( R_{ab}-\frac{1}{2} g_{ab}R+\nabla_{a} Z_{b}+\nabla_{b} Z_{a}-g_{ab}\nabla^c Z_c -\kappa_{1}[n_{a} Z_{b}+n_{b} Z_{a}+\kappa_{2} g_{ab} n_{c} Z^{c}] = 8\pi T_{ab} \)
Noabely, we transfer it into a first order formulation following
\( \partial_t Q +\nabla_i F_i(Q) +B_i (Q) \nabla_i Q = S(Q). \)
Collecting all derived quantities from CCZ4 decomposition, the gauge quantities and helper variables yields an evolving system of over 24 variables
\( Q(t) = \left( \tilde{\gamma}_{ij}, \alpha, \beta^i, \phi, \tilde{A}_{ij},K, \Theta, \hat{\Gamma}^i, b^i \right)(t), \)
if the auxiliary varibles representing the first-order derivatives:
\( A_{i}:=\partial_{i} \alpha, \ B_{k}^{i}:=\partial_{k} \beta^{i}, \ D_{k i j}:=\frac{1}{2} \partial_{k} \tilde{\gamma}_{i j}, \ P_{i}:=\partial_{i} \phi \)
are calculated on-fly. In this case we have the second-order formulation of the code. Adding the auxiliary variables to the system and treating them as further evolving variables make the code completely first-order, yielding a system of 58 independent variables with
\( Q(t) = \left( \tilde{\gamma}_{ij}, \alpha, \beta^i, \phi, \tilde{A}_{ij},K, \Theta, \hat{\Gamma}^i, b^i, A_k, B^i_k, D_{kij}, P_k \right). \)
Both formulations are now available in the code for difference scenarios. Users are referred to Appendix B of the paper below for more details of equations and formulation used in ExaGRyPE.
1.10.0