Divergence Correction by Modelling Numerical Errors

Errors

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

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

What I want to tell you about

• Correction in a simple example

? The problem

? Modelling errors

∗ Transport methods

∗ Generalised Lagrange multiplier methods

? The behaviour of the resulting error equations

What I want to tell you about

• Correction in a simple example

? The problem

? Modelling errors

∗ Transport methods

∗ Generalised Lagrange multiplier methods

? The behaviour of the resulting error equations

• What about Maxwell- and MHD-equations?

? Corrections in Maxwell equations

? How they come into MHD equations

? The Powell correction

Motivation

MHD equations:

ρ_t + ∇ · [ρ~v] = 0 , (ρ~v)_t + ∇ ·

ρ~v ·~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

Motivation

MHD equations:

ρ_t + ∇ · [ρ~v] = 0 , (ρ~v)_t + ∇ ·

ρ~v ·~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 .

A simple test Problem

A simple conservation law:

B_t = 0

A simple test Problem

A simple conservation law:

B_t = 0 A simple divergence constraint:

∇ · B = 0

A simple test Problem

A simple conservation law:

B_t = 0 A simple divergence constraint:

∇ · B = 0 Writing error terms into the equations:

B_t + err₁ = 0 err₂ + ∇ · B = 0

A simple test Problem

A simple conservation law:

B_t = 0 A simple divergence constraint:

∇ · B = 0 Writing error terms into the equations:

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

err_2t − ∇ · err₁ = 0

Modelling errors

What a reasonable error model has to fulfill:

• Simple form of error equations .

Modelling errors

What a reasonable error model has to fulfill:

• Simple form of error equations .

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

Modelling errors

What a reasonable error model has to fulfill:

• Simple form of error equations .

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

• Diminishing of errors locally and in L¹ norm.

Modelling errors

What a reasonable error model has to fulfill:

• Simple form of error equations .

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

• Diminishing of errors locally and in L¹ norm.

• Easy implementation of the resulting correction.

Modelling errors(2)

Possible forms of the error equation:

• transport equation

• wave equation

• heat equation

• Poisson equation

• telegraph equation

Transport methods

Model:

err₁ = −ψv˜ (˜v ∈ Rⁿ) , err₂ = ψ .

Error equation:

ψ_t + ∇ · (ψv) = 0˜

Implementation:

B_t = −(∇ · B)˜v

Characteristic solution of error equation

Quasi-linear form:

ψ_t + ˜v∇ψ = −(∇ · v)ψ˜

Characteristics:

˙

x(t) = ˜v(x(t), t)

Solution along characteristics:

ψ(x(t), t) = ψ₀e^{−R0t∇·v(x(s),s)˜ ds}

How to choose v ˜

• Should point to the outward on the boundary. (Diminish L¹-norm.)

How to choose v ˜

• Should point to the outward on the boundary. (Diminish L¹-norm.)

• Should have positive (at least nonnegative) Divergence. (Diminish locally.)

How to choose v ˜

• Should point to the outward on the boundary. (Diminish L¹-norm.)

• Should have positive (at least nonnegative) Divergence. (Diminish locally.) Examples:

Star-region with respect to the origin:

˜

v(x) = v₀ x

How to choose v ˜

• Should point to the outward on the boundary. (Diminish L¹-norm.)

• Should have positive (at least nonnegative) Divergence. (Diminish locally.) Examples:

Star-region with respect to the origin:

˜

v(x) = v₀ x

Higher dimensional analogue of an annulus ({x ∈ Rⁿ : 0 < r < kxk² < R}):

˜

v(x) =







v₀ x

1 − _kx^r˜_k2

if kxk² ≥ r˜ v_0kx^x_k

2

1

˜

r^k − _kx¹_kk 2

if kxk² < r˜ (r < r < R, k˜ ≥ n − 1)

One dimensional example

For v˜ = v₀x the solution for ψ is:

ψ(x, t) = ψ₀(x · e^−v0t)e^−v0t Special cases:

ψ₀(x) = x ⇒ ψ(x, t) = x · e^−2v0t . ψ₀(x) =

N

X

j=0

α_jx^j ⇒ ψ(x, t) =

N

X

j=0

α_jx^j e^−(j+1)v0t .

ψ₀(x) = 1

x ⇒ ψ(x, t) = 1

x .

Generalised Lagrange multiplier methods (GLM)

Model:

err₁ = ∇ψ ,

err₂ = g(ψ, ψ_t) . Error equation:

g(ψ, ψ_t)_t − ∆ψ = 0 .

Generalised Lagrange multiplier methods (GLM)

Model:

err₁ = ∇ψ ,

err₂ = g(ψ, ψ_t) . Error equation:

g(ψ, ψ_t)_t − ∆ψ = 0 . Evolution of the errors:

g(err₁,err_1t) − ∇(∇ · err₁) = 0 , g(err₂,err_2t) − ∆ err₂ = 0 .

Different forms of GLM-Correction

Hyperbolic correction: g(ψ, ψ_t) = 1

c²ψ_t → wave equation.

Different forms of GLM-Correction

Hyperbolic correction: g(ψ, ψ_t) = 1

c²ψ_t → wave equation.

Parabolic correction: g(ψ, ψ_t) = 1

κψ → heat equation.

Different forms of GLM-Correction

Hyperbolic correction: g(ψ, ψ_t) = 1

c²ψ_t → wave equation.

Parabolic correction: g(ψ, ψ_t) = 1

κψ → heat equation.

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

∇ψ as Lagrange multiplier.

Leads to projection method.

Different forms of GLM-Correction

Hyperbolic correction: g(ψ, ψ_t) = 1

c²ψ_t → wave equation.

Parabolic correction: g(ψ, ψ_t) = 1

κψ → heat equation.

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

∇ψ as Lagrange multiplier.

Leads to projection method.

Mixed type correction: g(ψ, ψ_t) = 1

c²ψ_t + 1

κψ → telegraph equation.

Properties of the different forms

Hyperbolic correction: Cheap in Computation. Conservation of errors (→ absor- bing boundary conditions).

Properties of the different forms

Hyperbolic correction: Cheap in Computation. Conservation of errors (→ absor- bing boundary conditions).

Parabolic correction: Restrictive time step condition for explicit computation.

High cost for implicit computation.

Properties of the different forms

Hyperbolic correction: Cheap in Computation. Conservation of errors (→ absor- bing boundary conditions).

Parabolic correction: Restrictive time step condition for explicit computation.

High cost for implicit computation.

Elliptic correction: Expensive in computation.

Properties of the different forms

Hyperbolic correction: Cheap in Computation. Conservation of errors (→ absor- bing boundary conditions).

Parabolic correction: Restrictive time step condition for explicit computation.

High cost for implicit computation.

Elliptic correction: Expensive in computation.

Mixed type correction: Cheap in computation. Errors not conserved (→ almost no care to be taken on boundary conditions). Our favourite!

Convergence results

Sonja Gutmann (2000):

For the GLM-corrected system of the hyperbolic, parabolic or mixed type with initial values

B₀ ∈ L²(Ω) , ψ₀ ≡ 0 and boundary values

• Homogeneous Dirichlet (parabolic correction).

• ψ − B · ~n = 0 (hyperbolic and mixed type correction).

the (weak) solution B converges to the divergence free part of B₀.

Travelling wave solutions

Ansatz:

err₂(x, t) = e^i(~kx−ωt) · e^αt , (α, ω ∈ R, ~k ∈ Rⁿ) Hyperbolic correction:

err₂ = e^{i(~kx±ck~kk2t)} . Parabolic correction:

err₂ = e^i~kx · e^−κ~k2t . Mixed type correction:

err₂ =









 eⁱ⁽

~kx±c r

~k²− ^c2

4κ2 t)

· e^−c

2

2κt if ~k² ≥ _4κ^c22 , e^i~kx · e⁻

c2 2κt±c

r

c2

4κ2−~k² t

if ~k² ≤ _4κ^c22 .

Corrections in the Maxwell equations

E_t − c²(∇ × B) + fehl₁ = − j ε₀ , B_t + (∇ × E) + err₁ = 0 ,

∇ · E + fehl₂ = q ε₀ ,

∇ · B + err₂ = 0 . Error equations:

fehl_2t − ∇ · fehl₁= 1

ε₀(q_t + ∇ · j) , err_2t − ∇ · err₁= 0 .

MHD equations deduced with corrected Maxwell equations

ρ_t + ∇ · [ρ~v] = 0 , (ρ~v)_t + ∇ ·

ρ~v ·~v^T + (p + 1

2B²)I − B · B^T

= fehl₁×B − (∇ · B)B , B_t + ∇ ·

~v · B^T − B ·~v^T

= −err₁ , e_t + ∇ · h

(e + p + 1

2B²)~v − B(~v · B)i

= −~v ·

fehl₁×B

−(~v · B)(∇ · B) − B·err₁ , fehl₂ − ∇ · [~v × B] = 0 ,

err₂ + ∇ · B = 0 .

MHD equations corrected directly

ρ_t + ∇ · [ρ~v] = 0 , (ρ~v)_t + ∇ ·

ρ~v · ~v^T + (p + 1

2B²)I − B · B^T

= −fehl₁ , B_t + ∇ ·

~v · B^T − B ·~v^T

= −err₁ , e_t + ∇ · h

(e + p + 1

2B²)~v − B(~v · B)i

= 0 , fehl₂ − ∇ · [~v × B] = 0 , err₂ + ∇ · B = 0 . Error equation for fehl involves most state variables.

Error equation for err same as for Maxwell and simple problem.

The Powell correction

ρ_t + ∇ · [ρ~v] = 0 , (ρ~v)_t + ∇ ·

ρ~v ·~v^T + (p + 1

2B²)I − B · B^T

= −(∇ · B)B , 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 With transport method for err and

˜

v = ~v .

Numerical Results for MHD

Divergence errors:

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

Numerical Results for MHD

Grid adaption:

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

Numerical Results for MHD

B-Field:

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

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.

Conclusions

• Error modelling gives insight in the known correction methods.

Conclusions

• Error modelling gives insight in the known correction methods.

• Error modelling is a platform for the development of better correction schemes.

Conclusions

• Error modelling gives insight in the known correction methods.

• Error modelling is a platform for the development of better correction schemes.

• The mixed type is the preferable GLM method.

Conclusions

• Error modelling gives insight in the known correction methods.

• Error modelling is a platform for the development of better correction schemes.

• The mixed type is the preferable GLM method.

• Divergence correction makes the scheme more stable.

Conclusions

• Error modelling gives insight in the known correction methods.

• Error modelling is a platform for the development of better correction schemes.

• The mixed type is the preferable GLM method.

• Divergence correction makes the scheme more stable.

• A good Divergence correction scheme saves computation time on adapted grids.

Conclusions

• Error modelling gives insight in the known correction methods.

• Error modelling is a platform for the development of better correction schemes.

• The mixed type is the preferable GLM method.

• Divergence correction makes the scheme more stable.

• A good Divergence correction scheme saves computation time on adapted grids.

• Good choice of parameters in GLM-methods?