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Simflowny is an open platform which automatically generates efficient parallel code of scientific dynamical models for different simulation frameworks. Here we present major upgrades on this software to support simultaneously a quite generic family of partial differential equations. These equations can be discretized using: (i) standard finite-difference for systems with derivatives up to any order, (ii) High-Resolution-Shock-Capturing methods to deal with shocks and discontinuities of balance law equations, and (iii) particle-based methods. We have improved the adaptive-mesh-refinement algorithms to preserve the convergence order of the numerical methods, which is a requirement for improving scalability. Finally, we have also extended our graphical user interface (GUI) to accommodate these and future families of equations. This paper summarizes the formal representation and implementation of these new families, providing several validation results., Comment: 11 pages, 6 figures, accepted in Computer Physics Communications

Simflowny is an open platform which automatically generates parallel code of scientific dynamical models for different simulation frameworks. Here we present major upgrades on this software to support an extended set of families of models, in particular: i) a new generic family for partial differential equations, which can include spatial derivatives of any order, ii) a new family for agent based models to study complex phenomena --either on a spatial domain or on a graph--. Additionally we introduce a flexible graphical user interface (GUI) to accommodate these and future families of equations. This paper describes the new GUI architecture and summarizes the formal representation and implementation of these new families, providing several validation results., Comment: 26 pages, 21 figures

The definition of complexity through Statistical Complexity Measures (SCM) has recently seen major improvements. Mostly, effort is concentrated in measures on time series. We propose a SCM definition for spatial dynamical systems. Our definition is in line with the trend to combine entropy with measures of structure (such as disequilibrium). We study the behaviour of our definition against the vectorial noise model of Collective Motion. From a global perspective, we show how our SCM is minimal at both the microscale and macroscale, while it reaches a maximum at the ranges that define the mesoscale in this model. From a local perspective, the SCM is minimum both in highly ordered and chaotic areas, while it reaches a maximum at the edges between such areas. These characteristics suggest this is a good candidate for detecting the mesoscale of arbitrary dynamical systems as well as regions where the complexity is maximal in such systems., Comment: 13 pages, 5 figures

A Lagrangian density is provided, that allows to recover the Z4 evolution system from an action principle. The resulting system is then strongly hyperbolic when supplemented by gauge conditions like '1+log' or 'freezing shift', suitable for numerical evolution. The physical constraint $Z_\mu = 0$ can be imposed just on the initial data. The corresponding canonical equations are also provided. This opens the door to analogous results for other numerical-relativity formalisms, like BSSN, that can be derived from Z4 by a symmetry-breaking procedure. The harmonic formulation can be easily recovered by a slight modification of the procedure. This provides a mechanism for deriving both the field evolution equations and the gauge conditions from the action principle, with a view on using symplectic integrators for a constraint-preserving numerical evolution. The gauge sources corresponding to the 'puncture gauge' conditions are identified in this context., Comment: Revised version, includes explicit expresions for gauge sources corresponding to 1+log and gamma-driver gauge conditions ('punctures' gauge)

A set of energy-momentum constraint-preserving boundary conditions is proposed for the first-order Z4 case. The stability of a simple numerical implementation is tested in the linear regime (robust stability test), both with the standard corner and vertex treatment and with a modified finite-differences stencil for boundary points which avoids corners and vertices even in cartesian-like grids. Moreover, the proposed boundary conditions are tested in a strong field scenario, the Gowdy waves metric, showing the expected rate of convergence. The accumulated amount of energy-momentum constraint violations is similar or even smaller than the one generated by either periodic or reflection conditions, which are exact in the Gowdy waves case. As a side theoretical result, a new symmetrizer is explicitly given, which extends the parametric domain of symmetric hyperbolicity for the Z4 formalism. The application of these results to first-order BSSN-like formalisms is also considered., Comment: Revised version, with conclusive numerical evidence. 23 pages, 12 figures

Gowdy waves, one of the standard 'apples with apples' tests, is proposed as a test-bed for constraint-preserving boundary conditions in the non-linear regime. As an illustration, energy-constraint preservation is separately tested in the Z4 framework. Both algebraic conditions, derived from energy estimates, and derivative conditions, deduced from the constraint-propagation system, are considered. The numerical errors at the boundary are of the same order than those at the interior points., Comment: 5 pages, 1 figure. Contribution to the Spanish Relativity Meeting 2009

The Osher-Chakrabarthy family of linear flux-modification schemes is considered. Improved lower bounds on the compression factors are provided, which suggest the viability of using the unlimited version. The LLF flux formula is combined with these schemes in order to obtain efficient finite-difference algorithms. The resulting schemes are applied to a battery of numerical tests, going from advection and Burgers equations to Euler and MHD equations, including the double Mach reflection and the Orszag-Tang 2D vortex problem. Total-variation-bounded behavior is evident in all cases, even with time-independent upper bounds. The proposed schemes, however, do not deal properly with compound shocks, arising from non-convex fluxes, as shown by Buckley-Leverett test simulations., Comment: Revised version, including new tests to appear in Journal of Computational Physics

The harmonic formulation of Einstein's field equations is considered, where the gauge conditions are introduced as dynamical constraints. The difference between the fully constrained approach (used in analytical approximations) and the free evolution one (used in most numerical approximations) is pointed out. As a generalization, quasi-stationary gauge conditions are also discussed, including numerical experiments with the gauge-waves testbed. The complementary 3+1 approach is also considered, where constraints are related instead with energy and momentum first integrals and the gauge must be provided separately. The relationship between the two formalisms is discussed in a more general framework (Z4 formalism). Different strategies in black hole simulations follow when introducing singularity avoidance as a requirement. More flexible quasi-stationary gauge conditions are proposed in this context, which can be seen as generalizations of the current 'freezing shift' prescriptions., Comment: Talk given at the Spanish Relativity Meeting, Tenerife, September 2007

An almost-stationary gauge condition is proposed with a view to Numerical Relativity applications. The time lines are defined as the integral curves of the timelike solutions of the harmonic almost-Killing equation. This vector equation is derived by a variational principle, by minimizing the deviations from isometry. The corresponding almost-stationary gauge condition allows one to put the field equations in hyperbolic form, both in the free-evolution ADM and in the Z4 formalisms., Comment: Talk presented at the Spanish Relativity Meeting, September 6-10 2005 Revised version

The constraint-preserving approach, which aim is to provide consistent boundary conditions for Numerical Relativity simulations, is discussed in parallel with other recent developments. The case of the Z4 system is considered, and constraint-preserving boundary conditions of the Sommerfeld type are provided. A necessary condition for the stability of the proposed boundary conditions is obtained, which amounts to the requirement of a symmetric ordering of space derivatives. This requirement is numerically seen to be also sufficient in the absence of corners and edges. Maximally dissipative boundary conditions are also implemented. In this case, a less restrictive stability condition is obtained, which is shown numerically to be also sufficient even in the presence of corners and edges., Comment: Enhanced version, including a new Appendix on maximally dissipative boundary conditions (12 pages, 5 figures)

The family of generalized-harmonic gauge conditions, which is currently used in Numerical Relativity for its singularity-avoidant behavior, is analyzed by looking for pathologies of the corresponding spacetime foliation. The appearance of genuine shocks, arising from the crossing of characteristic lines, is completely discarded. Runaway solutions, meaning that the lapse function can grow without bound at an accelerated rate, are instead predicted. Black Hole simulations are presented, showing spurious oscillations due to the well known slice stretching phenomenon. These oscillations are made to disappear by switching the numerical algorithm to a high-resolution shock-capturing one, of the kind currently used in Computational Fluid Dynamics. Even with these shock-capturing algorithms, runaway solutions are seen to appear and the resulting lapse blow-up is causing the simulations to crash. As a side result, a new method is proposed for obtaining regular initial data for Black Hole spacetimes, even inside the horizons., Comment: 8 pages, 9 figures

A class of dynamical shift conditions is shown to lead to a strongly hyperbolic evolution system, both in the Z4 and in the BSSN Numerical Relativity formalisms. This class generalizes the harmonic shift condition, where light speed is the only non-trivial characteristic speed, and it is contained into the multi-parameter family of minimal distortion shift conditions recently proposed by Lindblom and Scheel. The relationship with the analogous 'dynamical freezing' shift conditions used in black hole simulations discussed., Comment: text changed, accepted in PRD

The general-covariant Z4 formalism is further analyzed. The gauge conditions are generalized with a view to Numerical Relativity applications and the conditions for obtaining strongly hyperbolic evolution systems are given both at the first and the second order levels. A symmetry-breaking mechanism is proposed that allows one, when applied in a partial way, to recover previously proposed strongly hyperbolic formalisms, like the BSSN and the Bona-Mass\'o ones. When applied in its full form, the symmetry breaking mechanism allows one to recover the full five-parameter family of first order KST systems. Numerical codes based in the proposed formalisms are tested. A robust stability test is provided by evolving random noise data around Minkowski space-time. A strong field test is provided by the collapse of a periodic background of plane gravitational waves, as described by the Gowdy metric., Comment: figures added, text changed, accepted in PRD

A general covariant extension of Einstein\'{}s field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector $Z_\mu$. Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints amount to the covariant algebraic condition $Z_\mu=0$. The extended field equations can be supplemented by suitable coordinate conditions in order to provide symmetric hyperbolic evolution systems: this is actually the case for either harmonic coordinates or normal coordinates with harmonic slicing.

A general covariant extension of Einstein's field equations is considered with a view to Numerical Relativity applications. The basic variables are taken to be the metric tensor and an additional four-vector. The extended field equations, when supplemented by suitable coordinate conditions, determine the time evolution of all these variables without any constraint. Einstein's solutions are recovered when the additional four-vector vanishes, so that the energy and momentum constraints hold true. The extended system is well posed when using the natural extension of either harmonic coordinates or the harmonic slicing condition in normal coordinates, Comment: 3 pages, submitted to PRL

A suite of three evolution systems is presented in the framework of the 3+1 formalism. The first one is of second order in space derivatives and has the same causal structure of the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system for a suitable choice of parameters. The second one is the standard first order version of the first one and has the same causal structure of the Bona-Masso system for a given parameter choice. The third one is obtained from the second one by reducing the space of variables in such a way that the only modes that propagate with zero characteristic speed are the trivial ones. This last system has the same structure of the ones recently presented by Kidder, Scheel and Teukolski: the correspondence between both sets of parameters is explicitly given. The fact that the suite started with a system in which all the dynamical variables behave as tensors (contrary to what happens with BSSN system) allows one to keep the same parametrization when passing from one system to the next in the suite. The direct relationship between each parameter and a particular characteristic speed, which is quite evident in the second and the third systems, is a direct consequence of the manifest 3+1 covariance of the approach.

The gravitational radiation degrees of freedom of freedom are described in the framework of the 3+1 decomposition of spacetime. The relationship with eigenfields of the Kidder-Scheel-Teukolsky (KST) equations is established. This relationship is used to fix a parameter in the KST equations which is related to the ordering ambiguity of space derivatives in the Ricci tensor, which is inherent to first order evolution systems, like the ones currently used in Numerical Relativity applications., Comment: 4 pages, no figures, accepted in PRD

New numerical methods have been applied in relativity to obtain a numerical evolution of Einstein equations much more robust and stable. Starting from 3+1 formalism and with the evolution equations written as a FOFCH (first-order flux conservative hyperbolic) system, advanced numerical methods from CFD (Computational Fluid Dynamics) have been successfully applied. A flux limiter mechanism has been implemented in order to deal with steep gradients like the ones usually associated with black hole spacetimes. As a test bed, the method has been applied to 3D metrics describing propagation of nonlinear gauge waves. Results are compared with the ones obtained with standard methods, showing a great increase in both robustness and stability of the numerical algorithm., Comment: 9 pages, 5 figures. to be published in the Procedings of ERE01

A method for studying the causal structure of space-time evolution systems is presented. This method, based on a generalization of the well known Riemann problem, provides intrinsic results which can be interpreted from the geometrical point of view. A one-parameter family of hyperbolic evolution systems is presented and the physical relevance of their characteristic speeds and eigenfields is discussed. The two degrees of freedom corresponding to gravitational radiation are identified in an intrinsic way, independent of the space coordinate system. A covariant interpretation of these degrees of freedom is provided in terms of the geometry of the wave fronts. The requirement of a consistent geometrical interpretation of the gravitational radiation degrees of freedom is used to solve the ordering ambiguity that arises when obtaining first order evolution systems from the second order field equations. This achievement provides a benchmark which can be used to check both the existing and future first order hyperbolic formalisms for Numerical Relativity., Comment: 9 pages, no figures

The paper combines theoretical and applied ideas which have been previously considered separately into a single set of evolution equations for Numerical Relativity. New numerical ingredients are presented which avoid gauge pathologies and allow one to perform robust 3D calculations. The potential of the resulting numerical code is demonstrated by using the Schwarzschild black hole as a test-bed. Its evolution can be followed up to times greater than one hundred black hole masses., Comment: 11 pages, 4 figures; figure corrected

Instabilities in finite difference codes due to the singularity of spherical coordinates at the center are studied. In typical Numerical Relativity applications, standard regularization techniques by themselves do not ensure long term stability. A proposal to remedy that problem is presented, which takes advantage of redundant quantities introduced in recent hyperbolic formulations of Einstein's evolution equations. The results are discussed through the example case of a boson star, where a significant improvement in the implementation of boundary conditions is also presented., Comment: 10 pages, Latex, 1 postscript figure

Initial data corresponding to spacetimes containing black holes are considered in the time symmetric case. The solutions are obtained by matching across the apparent horizon different, conformally flat, spatial metrics. The exterior metric is the vacuum solution obtained by the well known conformal imaging method. The interior metric for every black hole is regular everywhere and corresponds to a positive energy density. The resulting matched solutions cover then the whole initial (Cauchy) hypersurface, without any singularity, and can be useful for numerical applications. The simpler cases of one black hole (Schwarzschild data) or two identical black holes (Misner data) are explicitly solved. A procedure for extending this construction to the multiple black hole case is also given, and it is shown to work for all time symmetric vacuum solutions obtained by the conformal imaging method. The numerical evolution of one such 'stuffed' black hole is compared with that of a pure vacuum or 'plain' black hole in the spherically symmetric case., Comment: 12 pages, Latex, 4 postscript figures, corrected some typos, new section about physical interpretation