Higher-order methods for relativistic magnetohydrodynamics

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Higher-order methods for relativistic magnetohydrodynamics

Jonatan Núñez-de la Rosa, Claus-Dieter Munz

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Higher-Order Methods for Relativistic Magnetohydrodynamics

Jonatan N´

u˜

nez-de la Rosa, Claus-Dieter Munz

Institut f¨

ur Aerodynamik und Gasdynamik, Universit¨

at Stuttgart

Contact: nunez@iag.uni-stuttgart.de

Higher-Order Methods for Relativistic Magnetohydrodynamics

Jonatan N´

u˜

nez-de la Rosa, Claus-Dieter Munz

Institut f¨

ur Aerodynamik und Gasdynamik, Universit¨

at Stuttgart

Contact: nunez@iag.uni-stuttgart.de

Abstract

A higher-order finite volume method based on WENO7 reconstruction for solving the relativistic magnetohydrodynamics equations in two-dimensional domains is pre-sented. In the presence of strong shocks, a WENO3 reconstruction is used instead. The time discretization is performed by a Strong Stability-Preserving Runge-Kutta method of fourth order. Numerical results include the Orszag-Tang vortex, the Rotor problem and the Spherical Blast Wave problem.

Relativistic Magnetohydrodynamics

Considering the Minkowski spacetime with Cartesian coordinates (t, x, y, z), the equations of the special relativistic magnetohydrodynamics form a system of con-servation laws which can be written as

∂ D

∂ t + ∇ · (D

v

) = 0, (Conservation of Mass) (1a)

∂

S

∂ t + ∇ · (

S

⊗

v

+ P) = 0, (Conservation of Momentum) (1b) ∂ E ∂ t + ∇ ·

S

= 0, (Conservation of Energy) (1c) ∂

B

∂ t + ∇ ·

B

⊗

v

−

v

⊗

B

= 0, (Induction Equations) (1d)

with the tensor P defined as P =  p + |

B

| 2 2Γ 2 + (

v

·

B

)2 2  I − 

B

Γ2 + (

v

·

B

)

v

‹ ⊗

B

, (2)

where the quantity in the first bracket is the total pressure (which has contributions from the thermal and magnetic pressure).

D = ρΓ , (3a)

S

= ρhΓ 2 + |

B

|2

v

− (

v

·

B

)

B

, (3b) E = ρhΓ 2 − p + |

B

| 2 2 + |

v

|2 |

B

|2 2 − (

v

·

B

)2 2 , (3c)

B

=

B

. (3d)

Γ is the Lorentz factor, defined by Γ = (1 − v2)−1/2.

Finite Volume Methods

Given the system of conservation laws

∂

u

∂ t + ∂

f

(

u

) ∂ x + ∂

g

(

u

) ∂ y = 0, (4)

by integrating it over the cell Ωi j, we get the semi-discrete scheme

d

u

_{i j} dt = ˆ

f

_i−1 2, j − ˆ

f

i+12, j ∆x + ˆ

g

_{i, j−}1 2, − ˆ

g

i, j+12, ∆y , (5)

where, in the context of finite volume methods,

u

_{i j} is the spatial average of

u

in the

cell Ωi j at time t

u

_{i j} = 1 ∆x 1 ∆y Z x_i +12 x_i −12 Z y_j +12 y_j −12

u

(x, y) dy dx (6) and ˆ

f

_i±1

2, j, and ˆ

g

i, j±12 are spatial averages of the physical fluxes over the cell faces

x_i±1 2, yj±12, respectively, at time t ˆ

f

_i±1 2, j = 1 ∆y N_{G P} X α=1

f u

(x_i±1 2, yα)
ωα,

g

ˆi, j±12 = 1 ∆x N_{G P} X α=1

g u

(x_α, y_j±1 2)
ωα. (7)

Because of only the cell averages

u

_{i j} are known, we require a high-order accurate

numerical procedure to reconstruct the point-wise values of

u

at the Gaussian

in-tegration points at the faces. By employing a high-order WENO reconstruction, we will have high-order accurate solutions in smooth parts of the flow, and essentially non-oscillatory properties around discontinuities.

Building Blocks for the RKFV

• Time Discretization . . . : SSPRK3, SSPRK4

• Spatial Reconstruction. . . .: WENO3, WENO5 and WENO7 • Numerical Fluxes . . . : Rusanov, HLL, HLLC, HLLD

• Solenoidal Constraint. . . .: Generalized Lagrange Multiplier • Shock Indicator . . . : Jameson Indicator

• Shock Capturing . . . : Robust WENO3 Scheme

Numerical Computations

Relativistic Current Sheet

0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 2.60 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90

Contour plots of the rest-mass density (left) and the square of the magnetic field magnitude (right) at simulation time

t = 7.5. A RKFV method has been used with WENO7

re-construction. A mesh of 600 × 600 cells has been employed.

Relativistic Orszag-Tang Vortex

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

Contour plots of the rest-mass density (left) and the square of the magnetic field magnitude (right) at simulation time

t = 1.0. A RKFV method has been used with WENO7

re-construction. A mesh of 600 × 600 cells has been employed.

Relativistic Spherical Blast Wave

1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 ·10−4 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60

Contour plots of the rest-mass density (left) and the Mach number (right) at simulation time t = 0.3. A RKFV method has been used with WENO7 reconstruction. A mesh of 600 × 600 cells has been employed.

Relativistic Rotor Problem

0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 0.00 0.50 1.00 1.50 2.00 2.50 3.00

Contour plots of the pressure (left) and the Mach number (right) at simulation time t = 0.4. A RKFV method has been used with WENO7 reconstruction. A mesh of 600 × 600 cells has been employed.

Relativistic Kelvin-Helmholtz Instability

1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 1.00 1.05 1.10 1.15 1.20 1.25 1.30 1.35 1.40 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 ·10−2 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 2.20 2.40 ·10−2