High Order Methods with Adaptive Mesh Refinement for the Solution of - PowerPoint PPT Presentation

High Order Methods with Adaptive Mesh Refinement for the Solution of the Relativistic MHD equations Olindo Zanotti University of Trento Collaborator: Michael Dumbser, University of Trento 14 th International Conference on Hyperbolic problems: Theory, Numerics, Applications. June 25 th 2012.

Outline • Part I: Astrophysical motivations for considering Resistive Relativistic MHD • Part II: The mathematics of Resistive Relativistic MHD: a hyperbolic system with stiff source terms • Part III: A numerical scheme for RRMHD • High order reconstruction • Using local space-time Discontinuous Galerkin for treating stiffness in the source terms • Construction of 1-step time evolution schemes • Adaptive Mesh Refinement • Part IV: High Lundquist number relativistic magnetic reconnection

Why Resistive MHD? In most cases, the ideal MHD approximation of infinite conductivity is correct: v L v L magnetic Reynolds and Lundquist number     0 A R 1, S 1 ratio between diffusion timescale and advection   m timescale In some cases, however, such an approximation is completely wrong and resistivity must be taken into account. This is particularly the case when magnetic reconnection takes place               2 B ( v B ) B 0 t       J B (Lyubarski 2005)

Where relativistic magnetic reconnection?  Giant flares in gamma-ray repeaters (Lyutikov 2003, 2006)  Current sheets at the Y point in pulsars magnetospheres  Dissipation of alternating fields at the termination shock of a relativistic pulsar wind (Petri & Lyubarski 2007)  Gamma-ray burst jets, where particle acceleration induced by magnetic reconnection is supposed to take place (Drenkhahn & Spruit 2002 , McKinney & Uzdensky 2010)  Accretion disc coronae of AGN where magnetic loops emerge from the disc via buoyancy instability (di Matteo 1998, Jaroschek 2004)

The Relativistic plasma • The energy-momentum tensor of a relativistic plasma is:                  T T T ( h u u pg ) F F ( F F ) g / 4   m em        2 2 2 2 g dx dx dt dx dy dz assume 3+1 formalism  • The Faraday EM tensor satisfies Maxwell’s equations              1 E 4 F J         E 4 , B J   c t c             F 0 1 B             F F / 2  B 0 , E   c t • Unlike ideal MHD, the current is not a derived quantity, and it is provided after specifying Ohm’s law                  J v [ E v B ( E v ) v ] c       • In ideal MHD one simply has E v B

Ohm’s law The electric conductivity is actually a tensor        J u e  e             ( g b b u b )    2      e e            , 1 , collision time     m m                  J q v [ E v B ( E v ) v ]                 2 ( E B )[ B v E ( B v ) v ]

RRMHD ideal RMHD         i i D ( Dv ) 0 D ( Dv ) 0 t i t i         i i S Z 0 S Z 0 t j i j t j i j         i i U S 0 U S 0 t i t i               i ijk i ijk B e E 0 B e E 0 t j k i t j k i           i ijk i i ijk E e B J E e v B t j k i j k                 i i B B t i t i          i E t i c     i q J 0 t i Hyperbolic Divergence Cleaning approach of Dedner et al. JCP, 175, 645, 2002 • where    D      i 2 i ijk S h v E B j k       2 2 2 U h p B / 2 E / 2            i 2 i i i 2 2 i Z h v v B B E E p B / 2 E / 2 j j j j j

Of course all of this holds also in curved spacetime   D W         2 S h W v E B 1      2 2 2 U h W p ( E B ) 2 + evolution equations for the electromagnetic field

RRMHD: a system with potential stiff source terms!  1 Timescale of advection       a u f ( u ) s ( u )   t x Timescale of dissipative process d               E B J t                 J q v [ E v B ( E v ) v ]    In the limit of the source becomes stiff   0     In the limit of the electric field evolves  1         2 B ( v B ) B 0 with a hyperbolic equation, not a parabolic one like in  t classical MHD Traditional approaches: 1. Strang-splitting, i.e. operator splitting, (Komissarov 2007, Zenitani et al. 2009) 2. Implicit-Explicit Runge Kutta (Palenzuela et al 2009)

Computational strategy 1. Apply a reconstruction operator to the Discontinuous Galerkin scheme at the beginning of each time step 2. Provide the time evolution of the reconstructed polynomials by solving the local space-time Discontinuous Galerkin predictor step 3. Solve the Riemann problem by some flux formula 4. Perform a one-step time update from time level n to n+1 with quadrature (or quadrature free) formulation

Reconstruction: the scheme (I) P N P M Set up a triangulation of space:    N x   ( m ) E x ( Q , )    ( m ) Q Q  m 1 Physical coordinates reference coordinates           ( m ) ( m ) ( m ) ( m ) ( m ) ( m ) ( m ) x X ( X X ) ( X X ) ( X X ) 1 2 1 3 1 4 1           ( m ) ( m ) ( m ) ( m ) ( m ) ( m ) ( m ) y Y ( Y Y ) ( Y Y ) ( Y Y ) 1 2 1 3 1 4 1           ( m ) ( m ) ( m ) ( m ) ( m ) ( m ) ( m ) z Z ( Z Z ) ( Z Z ) ( Z Z ) 1 2 1 3 1 4 1 4  z     , , [ 0 , 1 ] 3 4 1 y 2  3  1 x 2

Reconstruction: the scheme (II) P N P M At time t^n the vector of conserved quantities is represented by piecewise polynomials of degree N   L  N Traditional Galerkin     ˆ n n n u ( , t ) ( ) u ( t ) representation h l l  l 1 M  N From it, we reconstruct over polynomials of degree M :   L  M 1     ˆ     n n n w ( , t ) ( ) w ( t )  L M ( M 1 ) ( M d ) h l l d !  l 1 Number of degrees of freedom  Choose among Legendre polynomials l           d 0 , m , n , m n and coincide up to order N, while m n l l Q i

Reconstruction: the scheme (III) P N P M Choose a stencil for the reconstruction: The number of cells in the stencil is taken as n  e n L / L   ( m ) ( j ( k )) S T M N   n CEILING( L / L ) k 1 M N The reconstruction is obtained through L2 projection of the unknown polynomials Weak form of the identity of the reconstructed solution of degree M and the original numerical solution of degree N:            As many equations as the ( m ) w d u d Q S k h k h i element of the stencil. For stability Q Q reasons, the number of elements i i in the stencil is chosen larger than n Computed with Computed analytically Gaussian quadrature      ( m ) ( m ) w u for 1 l L l l N Solved using the constrained least squares technique

To recap : Reconstruction is applied to polynomials of degree N and generates polynomials of degree M Dumbser et al (2008) JCP, 227, 8209   N 0 classical FV   N M classical Galerkin   M N hybrid class