Conformal and covariant Z4 formulation of the Einstein equations: Strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes

Journal of Computational Physics

Fambri

Gaburro

et al. 2020

Self Cite

In this paper we propose an extension of the generalized Lagrangian multiplier method (GLM) of Munz et al. [52,30], which was originally conceived for the numerical solution of the Maxwell and MHD equations with divergence-type involutions, to the case of hyperbolic PDE systems with curl-type involutions.The key idea here is to solve an augmented PDE system, in which curl errors propagate away via a Maxwelltype evolution system. The new approach is first presented on a simple model problem, in order to explain the basic ideas. Subsequently, we apply it to a strongly hyperbolic first order reduction of the CCZ4 formulation (FO-CCZ4) of the Einstein field equations of general relativity, which is endowed with 11 curl constraints. Several numerical examples, including the long-time evolution of a stable neutron star in anti-Cowling approximation, are presented in order to show the obtained improvements with respect to the standard formulation without special treatment of the curl involution constraints.The main advantages of the proposed GLM approach are its complete independence of the underlying numerical scheme and grid topology and its easy implementation into existing computer codes. However, this flexibility comes at the price of needing to add for each curl involution one additional 3 vector plus another scalar in the augmented system for homogeneous curl constraints, and even two additional scalars for non-homogeneous curl involutions. For the FO-CCZ4 system with 11 homogeneous curl involutions, this means that additional 44 evolution quantities need to be added.Keywords: generalized Lagrangian multiplier approach (GLM), hyperbolic PDE systems with curl involutions, Einstein field equations with matter source terms, first order reduction of the CCZ4 system (FO-CCZ4), stable neutron star in anti-Cowling approximation

Section: Robust Stability Testmentioning

confidence: 99%

Section: Wavefield Generated By Two Rotating Massesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On GLM curl cleaning for a first order reduction of the CCZ4 formulation of the Einstein field equations

Journal of Computational Physics

Fambri

Gaburro

et al. 2020

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“…Finally, we presented a 3+1 split of the governing equations in Sec. 5.1 which are then solved using the ADER-DG family of high-order numerical schemes [14,15,16,38,37] designed specifically for hyperbolic partial differential equations, see Sec. 5.…”

Section: Numerical Examplesmentioning

confidence: 99%

A new continuum model for general relativistic viscous heat-conducting media

Romenski

Peshkov

Phil. Trans. R. Soc. A.

et al. 2020

The lack of formulation of macroscopic equations for irreversible dynamics of viscous heat-conducting media compatible with the causality principle of Einstein’s special relativity and the Euler–Lagrange structure of general relativity is a long-lasting problem. In this paper, we propose a possible solution to this problem in the framework of SHTC equations. The approach does not rely on postulates of equilibrium irreversible thermodynamics but treats irreversible processes from the non-equilibrium point of view. Thus, each transfer process is characterized by a characteristic velocity of perturbation propagation in the non-equilibrium state, as well as by an intrinsic time/length scale of the dissipative dynamics. The resulting system of governing equations is formulated as a first-order system of hyperbolic equations with relaxation-type irreversible terms. Via a formal asymptotic analysis, we demonstrate that classical transport coefficients such as viscosity, heat conductivity, etc., are recovered in leading terms of our theory as effective transport coefficients. Some numerical examples are presented in order to demonstrate the viability of the approach. This article is part of the theme issue ‘Fundamental aspects of nonequilibrium thermodynamics’.

“…Only after adding suitable first and second order ordering constraints, which arise from the definition of the auxiliary variables, it is possible to obtain a provably strongly hyperbolic and thus well-posed evolution system, denoted by FO-CCZ4 in the following. For all details of the derivation, the strong hyperbolicity proof and numerical results achieved with high order ADER-DG schemes, the reader is referred to [126]. In order to give an idea about the complexity of the Einstein field equations, it should be mentioned that one single evaluation of the FO-CCZ4 system requires about 20,000 floating point operations!…”

Section: A Strongly Hyperbolic First Order Reduction Of the Ccz4 Formmentioning

confidence: 99%

Efficient Implementation of ADER Discontinuous Galerkin Schemes for a Scalable Hyperbolic PDE Engine

Fambri

Tavelli

et al. 2018

Axioms

Self Cite

In this paper we discuss a new and very efficient implementation of high order accurate ADER discontinuous Galerkin (ADER-DG) finite element schemes on modern massively parallel supercomputers. The numerical methods apply to a very broad class of nonlinear systems of hyperbolic partial differential equations. ADER-DG schemes are by construction communication avoiding and cache blocking and are furthermore very well-suited for vectorization, so that they appear to be a good candidate for the future generation of exascale supercomputers. We introduce the numerical algorithm and show some applications to a set of hyperbolic equations with increasing level of complexity, ranging from the compressible Euler equations over the equations of linear elasticity and the unified Godunov-Peshkov-Romenski (GPR) model of continuum mechanics to general relativistic magnetohydrodynamics (GRMHD) and the Einstein field equations of general relativity. We present strong scaling results of the new ADER-DG schemes up to 180,000 CPU cores. To our knowledge, these are the largest runs ever carried out with high order ADER-DG schemes for nonlinear hyperbolic PDE systems. We also provide a detailed performance comparison with traditional Runge-Kutta DG schemes.The main challenge of nonlinear hyperbolic PDE arises from the fact that they can contain at the same time smooth solutions (like sound waves) as well as small scale structures (e.g. turbulent vortices), but they can also develop discontinuous solutions (shock waves) after finite times, even when starting from perfectly smooth initial data. These discontinuities were first discovered by Bernhard Riemann in his ground breaking work on the propagation of waves of finite amplitude in air [1,2], where the term finite should actually be understood in the sense of large, rather than simple sound waves of infinitesimal strength that have been considered in the times before Riemann. In the abstract of his work, Riemann stated that his discovery of the shock waves might probably not be of practical use for applied and experimental science, but should be mainly understood as a contribution to the theory of nonlinear partial differential equations. Several decades later, shock waves were also observed experimentally, thus confirming the new and groundbreaking mathematical concept of Riemann.The connection between symmetries and conservation laws were established in the work of Emmy Noether [3] at the beginning of the 20th century, while the first methods for the numerical solution of hyperbolic conservation laws go back to famous mathematicians such as . The connection between hyperbolic conservation laws, symmetric hyperbolic systems in the sense of Friedrichs [8] and thermodynamics was established for the first time by Godunov in 1961 [9] and was rediscovered again by Friedrichs and Lax in 1971 [10]. Within this theoretical framework of symmetric hyperbolic and thermodynamically compatible (SHTC) systems, established by Godunov and Romenski [11,12] it is possible to write down the Euler equ...