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Romatschke P.,Frankfurt Institute for Advanced Studies | Mendoza M.,ETH Zurich | Succi S.,Istituto per le Applicazioni Del Calcolo | Succi S.,Freiburg Institute for Advanced Studies
Physical Review C - Nuclear Physics

Starting from the Maxwell-Jüttner equilibrium distribution, we develop a relativistic lattice Boltzmann (LB) algorithm capable of handling ultrarelativistic systems with flat, but expanding, spacetimes. The algorithm is validated through simulations of a quark-gluon plasma, yielding excellent agreement with hydrodynamic simulations. The present scheme opens the possibility of transferring the recognized computational advantages of lattice kinetic theory to the context of both weakly and ultrarelativistic systems. © 2011 American Physical Society. Source

Mohseni F.,ETH Zurich | Mendoza M.,ETH Zurich | Succi S.,Istituto per le Applicazioni Del Calcolo | Succi S.,Freiburg Institute for Advanced Studies | And 2 more authors.
Physical Review D - Particles, Fields, Gravitation and Cosmology

We develop a relativistic lattice Boltzmann model capable of describing relativistic fluid dynamics at ultra-high velocities, with Lorentz factors up to γ∼10. To this purpose, we first build a new lattice kinetic scheme by expanding the Maxwell-Jüttner distribution function in an orthogonal basis of polynomials and applying an appropriate quadrature, providing the discrete versions of the relativistic Boltzmann equation and the equilibrium distribution. To achieve ultra-high velocities, we include a flux limiter scheme, and introduce the bulk viscosity by a suitable extension of the discrete relativistic Boltzmann equation. The model is validated by performing simulations of shock waves in viscous quark-gluon plasmas and comparing with existing models, finding very good agreement. To the best of our knowledge, we for the first time successfully simulate viscous shock waves in the highly relativistic regime. Moreover, we show that our model can also be used for near-inviscid flows even at very high velocities. Finally, as an astrophysical application, we simulate a relativistic shock wave, generated by, say, a supernova explosion, colliding with a massive interstellar cloud, e.g., molecular gas. © 2013 American Physical Society. Source

Mendoza M.,ETH Zurich | Karlin I.,ETH Zurich | Succi S.,Istituto per le Applicazioni Del Calcolo | Succi S.,Freiburg Institute for Advanced Studies | And 2 more authors.
Physical Review D - Particles, Fields, Gravitation and Cosmology

We develop a relativistic lattice Boltzmann (LB) model, providing a more accurate description of dissipative phenomena in relativistic hydrodynamics than previously available with existing LB schemes. The procedure applies to the ultrarelativistic regime, in which the kinetic energy (temperature) far exceeds the rest mass energy, although the extension to massive particles and/or low temperatures is conceptually straightforward. In order to improve the description of dissipative effects, the Maxwell-Jüttner distribution is expanded in a basis of orthonormal polynomials, so as to correctly recover the third-order moment of the distribution function. In addition, a time dilatation is also applied, in order to preserve the compatibility of the scheme with a Cartesian cubic lattice. To the purpose of comparing the present LB model with previous ones, the time transformation is also applied to a lattice model which recovers terms up to second order, namely up to the energy-momentum tensor. The approach is validated through quantitative comparison between the second- and third-order schemes with Boltzmann approach multiparton scattering (the solution of the full relativistic Boltzmann equation) for moderately high viscosity and velocities, and also with previous LB models in the literature. Excellent agreement with BAMPS and more accurate results than previous relativistic lattice Boltzmann models are reported. © 2013 American Physical Society. Source

Gosse L.,Istituto per le Applicazioni Del Calcolo
BIT Numerical Mathematics

The numerical approximation of one-dimensional relativistic Dirac wave equations is considered within the recent framework consisting in deriving local scattering matrices at each interface of the uniform Cartesian computational grid. For a Courant number equal to unity, it is rigorously shown that such a discretization preserves exactly the $$L^2$$L2 norm despite being explicit in time. This construction is well-suited for particles for which the reference velocity is of the order of $$c$$c, the speed of light. Moreover, when $$c$$c diverges, that is to say, for slow particles (the characteristic scale of the motion is non-relativistic), Dirac equations are naturally written so as to let a “diffusive limit” emerge numerically, like for discrete 2-velocity kinetic models. It is shown that an asymptotic-preserving scheme can be deduced from the aforementioned well-balanced one, with the following properties: it yields unconditionally a classical Schrödinger equation for free particles, but it handles the more intricate case with an external potential only conditionally (the grid should be such that $$c \Delta x\rightarrow 0$$cΔx→0). Such a stringent restriction on the computational grid can be circumvented easily in order to derive a seemingly original Schrödinger scheme still containing tiny relativistic features. Numerical tests (on both linear and nonlinear equations) are displayed. © 2014, Springer Science+Business Media Dordrecht. Source

Gosse L.,Istituto per le Applicazioni Del Calcolo
Journal of Computational and Theoretical Transport

Well-balanced schemes for two types of (1 + 1)-dimensional kinetic models are set up relying on scattering matrices derived from the explicit so–called elementary solutions of the corresponding stationary equations. A “matrix balancing” (or “matrix equilibration”) technique is evoked to ensure the overall mass (and positivity) preservation. The first model is a simplification of edge plasma dynamics close to a wall, like a tokamak divertor, whereas the second one is essentially a rewriting of the Anderson-Witting Bhatnagar-Gross-Krook (BGK) model of relativistic Boltzmann equation for example, photons. Numerical results on coarse grids are provided to illustrate the feasibility of the algorithms. 2016 © Taylor & Francis Group, LLC Source

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