A general approach to divergence correction in MHD

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MHD

Friedemann Kemm IAG Universit¨at Stuttgart kemm@iag.uni-stuttgart.de

Claus-Dieter Munz IAG Universit¨at Stuttgart munz@iag.uni-stuttgart.de

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Motivation

MHD equations:

ρ_t + ∇ · [ρv] = 0 (ρv)_t + ∇ ·

ρv · v^T + (p + 1

2B²)I − B · B^T

= 0 B_t + ∇ ·

v · B^T − B · v^T

= 0 e_t + ∇ · h

(e + p + 1

2B²)v − B(v · B) i

= 0

with divergence constraint

∇ · B = 0

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Motivation

MHD equations:

ρ_t + ∇ · [ρv] = 0 (ρv)_t + ∇ ·

ρv · v^T + (p + 1

2B²)I − B · B^T

= 0 B_t + ∇ ·

v · B^T − B · v^T

= 0 e_t + ∇ · h

(e + p + 1

2B²)v − B(v · B) i

= 0

with divergence constraint

∇ · B = 0

Complicated! → Skip all terms without influence on ∇ · B-problem

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A simplified system

The system:

B_t = 0

∇ · B = 0

Over-determined!

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A simplified system

The system:

B_t = 0

∇ · B = 0

Over-determined!

→ Writing artificial error terms into the equations:

B_t + err₁ = 0 err₂ + ∇ · B = 0 Resulting error equation:

err₂_t − ∇ · err₁ = 0

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Modelling the artificial error terms

What a reasonable error model has to fulfill:

• Simple form for error equations

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Modelling the artificial error terms

• Only one new scalar variable (Otherwise under-determined)

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Modelling the artificial error terms

• Diminish errors locally (L^∞ norm) and globally (L¹ norm)

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Modelling the artificial error terms

• Diminish errors locally (L^∞ norm) and globally (L¹ norm)

• Easy implementation of the resulting correction

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Two general possibilities

1. Algebraic expressions to model the artificial error terms

→ Transport Methods

2. Differential expressions to model the artificial error terms

→ Generalised Lagrange multiplier methods (GLM)

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Transport methods

Model:

err₁ = −ψv˜ (˜v ∈ Rⁿ) err₂ = ψ

Error equation:

ψ_t + ∇ · (ψv) = 0˜ Implementation:

B_t = −(∇ · B)˜v

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Properties of the error equation

Quasi-linear form:

ψ_t + ˜v∇ψ = −(∇ · v)ψ˜

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Properties of the error equation

Quasi-linear form:

ψ_t + ˜v∇ψ = −(∇ · v)ψ˜

Therefore v˜ should

• be directed to the outward on the boundary (diminish globally)

• have positive (at least nonnegative) divergence (diminish locally)

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Generalised Lagrange multiplier methods (GLM)

Model:

err₁ = ∇ψ

err₂ = g(ψ, ψ_t) Error equation:

g(ψ, ψ_t)_t − ∆ψ = 0

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Different forms of GLM-Correction

g(ψ, ψ_t)_t − ∆ψ = 0

Hyperbolic correction: g(ψ, ψ_t) = 1

c²ψ_t → wave equation

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Different forms of GLM-Correction

g(ψ, ψ_t)_t − ∆ψ = 0

c²ψ_t → wave equation Parabolic correction: g(ψ, ψ_t) = 1

κψ → heat equation

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Different forms of GLM-Correction

g(ψ, ψ_t)_t − ∆ψ = 0

Elliptic correction: g(ψ, ψ_t) ≡ 0 → Poisson equation

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Different forms of GLM-Correction

g(ψ, ψ_t)_t − ∆ψ = 0

Elliptic correction: g(ψ, ψ_t) ≡ 0 → Poisson equation

Mixed type correction: g(ψ, ψ_t) = 1

c²ψ_t + 1

κψ → telegraph equation

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Properties of the different forms

Hyperbolic correction: Cheap in Computation Conservation of errors Consistent with physics

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Properties of the different forms

Parabolic correction: Restrictive time step condition for explicit computation High cost for implicit computation

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Properties of the different forms

Elliptic correction: Expensive in computation

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Properties of the different forms

Elliptic correction: Expensive in computation Mixed type correction: Cheap in computation

Errors damped (not conserved) No resonance effects

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Back to the full MHD-system

ρ_t + ∇ · [ρv] = 0 (ρv)_t + ∇ ·

ρv · v^T + (p + 1

2B²)I − B · B^T

= 0 B_t + ∇ ·

v · B^T − B · v^T

= −err₁ e_t + ∇ · h

(e + p + 1

2B²)v − B(v · B) i

= 0 err₂ + ∇ · B = 0

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Back to the full MHD-system

ρ_t + ∇ · [ρv] = 0 (ρv)_t + ∇ ·

ρv · v^T + (p + 1

2B²)I − B · B^T

= 0 B_t + ∇ ·

v · B^T − B · v^T

= −err₁ e_t + ∇ · h

(e + p + 1

2B²)v − B(v · B) i

= 0 err₂ + ∇ · B = 0

Transport method with v˜ = v → Powell correction

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Numerical implementation

Transport method:

Like Powell correction with v replaced by artificial v˜ in the right hand side term

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Numerical implementation

Transport method:

GLM-Methods:

Elliptic: Projection method.

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Numerical implementation

Transport method:

GLM-Methods:

Parabolic: Substitute ∇ · B-equation in evolution of B Solve like usual viscosity term

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Numerical implementation

Transport method:

GLM-Methods:

Hyperbolic: Operator splitting: Corrected test system as predictor and standard MHD as corrector

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Numerical implementation

Transport method:

GLM-Methods:

Hyperbolic: Operator splitting: Corrected test system as predictor and standard MHD as corrector

Mixed: Like hyperbolic; just multiply ψ by e⁻^c

2

κ ∆t after each time step

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The 2d Riemann problem

0

1 4 2

3

^x

y

1 → 2: rarefaction wave

2 → 3: downward running shock 3 → 4: left running shock

4 → 1: general RP with several waves

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Numerical Results for MHD

B-Field:

B₁-component: a) Powell, b) GLM B₂-component: a) Powell, b) GLM

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Numerical Results for MHD

Divergence errors:

a) without correction, b) Powell correction, c) GLM correction

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Numerical Results for MHD

Grid adaption:

a) without correction, b) Powell correction, c) GLM correction

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Conclusions

• Error modelling gives insight into the known correction methods

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Conclusions

• Error modelling is a platform for development of improved correction schemes

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Conclusions

• Powell correction can be enhanced using an artificial velocity

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Conclusions

• Optimised divergence correction scheme saves computation time on adapted grids

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Conclusions

• Best choice of parameters in GLM-methods?

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Conclusions

• Quasi neutrality?

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Conclusions

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Conclusions

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Conclusions

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Numerical Results for MHD

Norm of divergence error:

0 2 4 6 8 10 12

0 0.05 0.1 0.15 0.2 0.25

L1(|divBjmp|)

time cp = 2.0

cp = 10.0 Powell no correction

0 200 400 600 800 1000 1200 1400 1600 1800

0 0.05 0.1 0.15 0.2 0.25

max(|divBjmp|)

time cp = 2.0

cp = 10.0 Powell no correction

L¹ norm, sup norm.