Approximate Harten-Lax-Van Leer (HLL) Riemann Solvers for Relativistic hydrodynamics and MHD Andrea Mignone Collaborators: G. Bodo, M. Ugliano Dipartimento di Fisica Generale, Universita di Torino (Italy) INAF Osservatorio Astronomico di Torino (Italy)
Outline Equations and numerical formalism; Riemann problem for relativistic hydro and MHD HLL / HLLC /HLLD approximate Riemann solvers; Performance and Astrophysical Applications Summary
Fundamental Equations Conservation of mass and energy-momentum for an ideal relativistic magnetized fluid in a special relativity 1 : Equations of State (EoS) provides closure w=w(,p); Nonlinear system of 8 hyperbolic conservation laws stating conservation of mass, momentum-energy + Maxwell s equations + B = 0 constraint. allow propagation of 7 waves: fast, slow, Alfvèn, entropy; 1 Lichnerowicz 1967, Anile 1990
Numerical approach Hyperbolic Conservation Laws: + Conserved quantities: Flux tensor Conservative discretization: F,G,H: time-averaged fluxes computed from the solution of Riemann problems;
1D Flux Computation: Riemann Problem In 1D, integrate over space (Δx) and time (Δt) Computation of the flux requires the (exact or approximate) solution of the Riemann problem at zone edges; Riemann Problem: given left and right states at a zone edge answer: the solution depends on the form of the conservation law.
1 st Order Godunov Formalism Solve riemann problem here u(x) Start with zone averaged values: x Solve riemann problem Compute fluxes
Riemann Problem in Relativistic Hydro t c entropy L acoustic R acoustic U L left state U R, right state x 3 wave pattern: left, middle and right wave Rankine-Hugoniot jump conditions, for each wave Across the contact wave only density has a jump, [v x ] = [p gas ]=0.
Riemann Problem in Relativistic MHD t Alfven slow [S/R] entropy slow [S/R] Alfven fast [S/R] Fast [S/R] U L, left state U R, right state x 7 wave pattern, across the contact wave, for B n 0, only density has a jump; across Alfven waves, [ ] = [p gas ]=0 but normal velocity [v x ] 0 magnetic field elliptically polarized.
An example
Riemann Solvers: different approaches Exact Riemann solver (nonlinear) Full nonlinear solution: Giacomazzo & Rezzolla, JFM (2006) Impracticable for heavily usage in upwind codes; Linearized Riemann solvers (Roe type) require characteristic decomposition in left and righteigenvectors may be prone to numerical pathologies [Komissarov, MNRAS (1999); Balsara, ApJS (2001); Koldoba MNRAS (2002); Anton et al, ApJS (2010)] HLL-type Riemann solvers (guess-based) based on guess to the signal speeds and on the integral average of the solution over the Riemann Fan; preserve positivity HD/MHD [Harten, Lax, van Leer, SIAM Rev.(1983); Toro (1997); Li, JCP (2005); Miyoshi & Kusano, JCP(2005) ] RHD/RMHD [Del Zanna et al, A&A (2003), Mignone et al, MNRAS (2005,2006,2009); Honkkila & Janhunen, JCP (2007)]
Approximate Schemes:HLLE L fast F *, U * R fast HLL (or HLLE): The Harten Laxvan Leer is based on a 2 wave approximation of the Riemann Fan. Remaining structure lumped into a single state. Requires: guess to outermost waves [Del Zanna et al, A&A (2003)] U L U R 2N jump conditions, 2N unknowns,!! F * F(U * )
Approximate schemes: HLLC L fast c contact HLLC: restore the middle contact (or tangential mode). F L*, U L * F L*, U L * R fast Riemann fan approximated by 3 waves. Requires: guess to outermost waves, closed form solution. U L U R Mignone & Bodo, MNRAS (2005, 2006); Honkkila & Janhunen, JCP (2007) 2N jump conditions + m additional constraints at middle wave; 4N+1 unknowns undetermined system (not consistent); F* and U* can no longer be considered completely independent; Further assumptions have to be made on the form of the fluxes.
HLLC for Relativistic Hydro L fast c contact 1D Relativistic hydro equations: F L*, U L * R fast F L*, U L * U L U R Express U* and F* in terms of a common set of 6 variables 5 PDEs; 10 jumps + 3 constraints at the middle wave: [p]=[v x ]=0, c = v x ; Need 10+3=13 unknowns: 6 (U * L) + 6 (U * R) + c Write down the jump conditions explicitly Solution a 2 + b + c = 0
Performance Results HLLC improves over HLL attaining sharper representation of discontinuities 1 1 st order scheme 2 nd order scheme 1 Mignone & Bodo, MNRAS(2005)
HLLC for Relativistic MHD L fast F L*, U L * c tangential F L*, U L * R fast HLLC extended to RMHD [Mignone & Bodo, MNRAS (2006), Honkkila & Janhunen, JCP (2007)] Solution vector now has 7 components (B x constant); 14 jumps + 6 constraints at the middle wave: U L U R Express U* and F* in terms of a common set of 10 variables Need 20 unknowns: 10 (U * L) + 10 (U * R) Solution a 2 + b + c = 0
Performance Results 1D shock tube 2D Shock Cloud Interaction
Performance Results Axisymmetric propagation of relativistic jet carrying toroidal field Choice of solver may give slower convergence rate
HLLD for Relativistic MHD L fast al Alfven c tangential ar Alfven R fast Riemann fan approximated by 5 waves; restore Alfven + contact modes. 14 jumps + 6 constraints (tangential c ) 12 constraints (Alfven a ) U L U R Express U* and F* in terms of a common set of 8 variables Only total pressure is constant through the Riemann fan. Need 32 unknowns Solution f(p) = 0 root of one nonlinear scalar equation in the the total pressure (no closed form) 1 1- Mignone et al. MNRAS(2009)
Performance Results Relativistic Brio-Wu Shock Tube Colliding relativistic stream
Applications to Kelvin-Helmholtz flows Linear and nonlinear evolution of a magnetized relativisitc shear layer; Computations at Low(L), Medium(M) and High (H) resolutions shows similar growth rates for HLLD, slower convergence for HLL:
Dissipation properties Decay of turbulent magnetic field ( 0.4) in a relativistic ( 4) shear flow; Average magnetic energy vs. time at different resolutions and Riemann solver: HLL requires a factor 2 in resolution to achieve comparable dissipation level
Axisymmetric Jet propagation Turbulent Jet Layers
CPU Time vs Accuracy HLLD slower than HLL by 10-90 % (problem dependent); HLLD HLL may need up to twice the resolution to achieve same accuracy / dissipation level; In D dimensions, T CPU (N) N D+1 HLL In 3D, for smooth flows, HLLD can gain up to a factor of 8 More effective in triggering small wavelength modes; More convenient to use expensive solver on fewer grid zones!
Relativistic MHD jets: 2D vs 3D 3D simulations confirm that field topology is essential in determining the dynamics 1 ; Jets carrying dominant toroidal field unstable to CD kink modes. Axisymmetric ( 2.5 D) Fully 3-D 1 Mignone et al. (MNRAS, 2010)
Application to Astrophysics: 3D Jet HLLD successfully applied to 3D Relativistic MHD simulations of Jets; [Mignone et al, MNRAS(2010)] - 640x1600x640 grid points using the PLUTO code 1 - Current-Driven instabilities lead to jet wiggling and deflection; - Shielding of inner core from interaction with environment prevent loss of momentum transfer. http://plutocode.ph.unito.it
Morphological Comparison Hydro Poloidal Toroidal secondary shocks Morphology of toroidal dominated field more similar to FR II sources
Summary Riemann solvers fundamental blocks for shock-capturing Godunov-type schemes; Level of accuracy/dissiption number of waves included in the approximate solutions: HLL: 2 waves, linear system HLLC: 3 waves, quadratic equation HLLD: 5 waves, nonlinear scalar equation Extension to multidimensional computations straightforward; Using more accurate solver with fewer points more efficient than diffusive solver with lots of points: Accuracy, Convergence properties, dissipation and small scale structures All simulations performed with PLUTO Code http://plutocode.ph.unito.it
Thank You