Discontinuous Galerkin methods for magnetohydrodynamics

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Discontinuous Galerkin methods for magnetohydrodynamics

Jonatan Núñez-de la Rosa, Claus-Dieter Munz

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Discontinuous Galerkin Methods for 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

Discontinuous Galerkin Methods for 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 discontinuous Galerkin spectral element method (DGSEM) for solv-ing the magnetohydrodynamics equations in two-dimensional domains is presented. A novel hybrid DGSEM/FV is used as shock capturing strategy. Those cells marked as troubled are evolved with a robust Finite Volume method with WENO3 recon-struction, and the nodal DG values in the cell are interpreted as FV subcell values. A Strong Stability-Preserving Runge-Kutta (SSPRK) method is used for the time dis-cretization. Numerical results include the Orszag-Tang Vortex, the Spherical Blast Wave problem and the Kelvin-Helmholtz instability.

Magnetohydrodynamics

The equations of the magnetohydrodynamics model the physical phenomena involv-ing electrically conductinvolv-ing fluid flow in which the electromagnetic forces can be of the same order or even greater than hydrodynamic ones. They are given by the con-servation of mass, the concon-servation of momentum, the concon-servation of energy and induction equations, together with the evolution equation of the scalar ψ from the divergence cleaning by Dedner et al. (2002):

∂ ρ

∂ t + ∇ ·

S

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

∂

S

∂ t + ∇ ·

S

⊗

v

+ P
= 0, (Conservation of Momentum) (1b)

∂ E

∂ t + ∇ · E

v

+ P ·

v

= 0, (Conservation of Energy) (1c)

∂

B

∂ t + ∇ ·

B

⊗

v

−

v

⊗

B

+ ψI
= 0, (Induction Equations) (1d)

∂ ψ ∂ t + ∇ · c 2 h

B

= − c_h2 c2 p

ψ. (Divergence Constraint) (1e)

We deal with the divergence constraint by using the Generalized Lagrange Multiplier method of Dedner et al. (2002). The pressure tensor appearing in equations (1b) and (1c) combines the influence of the hydrodynamic and the magnetic pressure (the quantity in brakets is the total pressure)

P = 

p + 1₂ |

B

_|2

‹

I −

B

⊗

B

. (2)

An equation of state (EOS) is used to close the system. In this work we make use of the ideal gas equation of state with adiabatic exponent γ

p = (γ − 1) E − 1 2ρ |

v

| 2 − 1₂ |

B

_|2 ‹ . (3)

Discontinuous Galerkin Methods

Given the system of conservation laws

∂

u

∂ t + ∇x ·

f

(

u

) = 0, (4)

where

u

=

u

(

x

, t) is the vector of conserved quantities, and

f

= 

f

₁,

f

₂,

f

₃ the

tensor of fluxes. The mapping between the physical and the reference element is

defined as

x

=

X

(ξ). The conservation law in the reference element takes the form

∂

u

∂ t +

1

J ∇ξ · ˜

f

= 0. (5)

And the weak formulation of the equation (5) is Z

E

J (ξ)∂

u

∂ t φ(ξ) dξ

| {z }

Time Derivative Integral

− Z E ˜

f

_{· ∇ξφ}(ξ) dξ | {z } Volume Integral + I ∂ E 

f

_{· ˆ}

n

∗σφ(ξ) dσ | {z } Surface Integral = 0. (6)

In every hexahedral element, we approximate the vector of conserved variables and the contravariant fluxes by polynomials, which basis are tensor products of

one-dimensional Lagrange polynomials of degree N: ψ_{i jk(ξ) = `i(ξ</sub>1<sub>)`j(ξ}2_)`k(ξ3_{). The}

semi-discrete formulation has the following form d ˆ

u

_{i jk} dt = − 1 J_{i jk}  N X λ=0 ˆD_iλ

f

ˆ_{λ jk}1 + €

f

∗σ+ξ 1 jk ˆ`i(+1) + 

f

∗σ−ξ 1 jk ˆ`i(−1) Š  − 1 J_{i jk}  N X µ=0 ˆD_jµ

_f

ˆ2 iµk + €

f

∗σ+ξ 2 ik ˆ`j(+1) + 

f

∗σ−ξ 2 ik ˆ`j(−1) Š  − 1 J_{i jk}  N X ν=0 ˆD_kν

_f

ˆ3 i jν + €

f

∗σ+ξ 3 i j ˆ`k(+1) + 

f

∗σ−ξ 3 i j ˆ`k(−1) Š  . (7)

Building Blocks for the RKDG

• Time Discretization . . . : SSPRK3, SSPRK4 • Spatial Discretization . . . : DGSEM with N ≥ 5

• Numerical Fluxes . . . : Rusanov, HLL, HLLC, HLLD

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

• Shock Capturing . . . : Hybrid DG/FV with WENO3 • Parallelization . . . : MPI

• Output Data Format . . . : HDF5

• Data Postprocessing . . . : TECPLOT, PARAVIEW

Numerical Computations

One-dimensional Riemann Problem

0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 x ρ EXACT DGSEM 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x By EXACT DGSEM

Defining u = ρ, v_x, v_y, v_z, p, B_x, B_y, B_z
. The initial

con-dition is determined by the following left and right states:

u_L = 1, 0, 0, 0, 1, 1, 1, 0
and u_R = 0.2, 0, 0, 0, 0.1, 1, 0, 0
.

The solution at t = 0.15 was obtained with DGSEM with

N = 9 and nElemX = 100, and SSPRK4. Hybrid DG/FV as

shock capturing, with WENO3 reconstruction.

Current Sheet 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75 1.40 1.45 1.50 1.55 1.60 1.65 1.70 1.75

Contour plots of the magnetic pressure at simulation times

t = 1.5 and t = 5.5. A RKDG method has been used with N =

5 and a mesh of 160 × 160 cells has been employed. Hybrid DG/FV as shock capturing, with WENO3 reconstruction.

Orszag-Tang Vortex 3.00 4.00 5.00 6.00 3.00 4.00 5.00 6.00

Contour plots of the density at simulation time t = 0.5. A RKDG method has been used with N = 5 and a mesh of 160 × 160 cells has been employed. Hybrid DG/FV as shock capturing, with WENO3 reconstruction.

Spherical Blast Wave

0.20 0.40 0.60 0.80 1.00 0.20 0.40 0.60 0.80 1.00

Contour plots of the pressure at simulation time t = 0.25. A RKDG method has been used with N = 5 and a mesh of 160 × 160 cells has been employed. Hybrid DG/FV as shock capturing, with WENO3 reconstruction.

Kelvin-Helmholtz Instability 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00