A Carbuncle-free Roe-Type Solver for the Euler Equations

Equations

Friedemann Kemm BTU Cottbus

kemm@math.tu-cottbus.de

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Stability of Discrete Shock Profiles

1d:

• Post-shock oscillations (Quirk 1994; Jin & Liu 1996; Arora & Roe 1997, . . . )

• Godunov scheme: unstable discrete profiles (Bultelle, Grassin, Serre 1998)

⇒ tend to neighbouring stable profiles

2d:

• High resolution Riemann solvers produce unstable profiles (Dumbser, Moschetta, Gressier 2004)

• Same mechanism in Carbuncle and Odd-Even-Decoupling (Chauvat, Moschetta, Gressier 2005)

1d-Stability ↔ 2d-Stability

entropy

transport jump backwards

jump forward

original shock location

y

x

shear

wave

⇒ Stabilization by viscosity on linear waves parallel to shock front

HLLE Solver

x t

S_l S_r

q_l q_r

q_HLL

• HLL: Constant intermediate state according to conservation

• HLLE: Natural choice of bounding speeds:

S_L = min{u˜ − a, u˜ _l − a_l,0} S_R = max{u˜ + ˜a, u_r + a_r,0}

• High viscosity on shear and entropy waves

HLLEM

• Comparison of viscosity matrices

• With Roe eigenvalues as S_L and S_R

g_Roe(q_r,q_l) = g_HLL(q_r,q_l) − u˜² − ˜a²

4˜a κ[˜l^T₂ ∆q˜r₂ + ˜l^T₃ ∆q˜r₃].

with

κ = 2˜a

˜

a + |u|˜

• Now for g_HLL take S_L, S_R like for HLLE → HLLEM.

• Exact resolution of entropy- and shear waves (Park, Kwon 2002)

HLL as Modification of Roe

x

˜ t

u −a˜ u˜ − φ(θ) ˜a u˜ u˜ +φ(θ) ˜a u˜ + ˜a

• Harten Hyman type splitting of contact wave

• HLL for φ(θ) = 1

• Same flux with HLLEM and κ replaced by (1 − φ(θ))κ

Desirable Properties of the new Solver

• Exact Resolution of single discontinuities

• No carbuncle

• No information from neighbouring Riemann problems needed

flux to compute

strong shock?

strong shock?

Indicator for Entropy- and Shear Waves

Rankine-Hugoniot condition for single contact or shear wave:

f(q_r) − f(q_l) = ˜u(q_r − q_l)

Idea: Residual in Rankine-Hugoniot condition as indicator:

R := f(q_r) − f(q_l) − u(q˜ _r − q_l)

Relate to flow magnitudes:

θ =

R

˜ a

2

Completing the Switching Function

Viscosity bounded by HLL(E):

φ(θ) = min{1, θ}

Relax by some parameter:

φ(θ) = min{1, ε θ}

Less dangerous when flow component parallel to shock:

φ(θ) = min{1, ε θ max{0,1 − M_u^α}} , α > 0

Make φ concave (experimental):

φ(θ) = min{1,(ε θ max{0,1 − M_u^α})^β} , 0 < β < 1

Application of the Switch

Roe:

• Split wave with u˜ into waves with u˜ − φ(θ)˜a and u˜ + φ(θ)˜a ⇒ RoeCC HLLEM:

• Multiply anti-diffusion coefficient κ by 1 − φ(θ) ⇒ HLLEMCC Both fluxes identical apart from entropy fix

Reasonable setting:

ε = 1

100 , α = β = 1 3

Quirk Test

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Quirk test: Godunov

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Quirk test: HLLEMCC, eps=0.01

Godunov HLLEMCC

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Quirk test: HLLEM

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Quirk test: HLLE

HLLEM HLLE

Steady Shock

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steady shock, Godunov

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steady shock, HLLEMCC

Godunov HLLEMCC

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steady shock, HLLEM

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steady shock, HLLE

HLLEM HLLE

Colliding Flow (2nd-Order)

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Colliding flow: Godunov

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Colliding flow: HLLEMCC

Godunov HLLEMCC

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Colliding flow: HLLEM

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Colliding flow: HLLE

HLLEM HLLE

Colliding Flow (2nd-Order)

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0 5 10 15 20 25 30 0 10 20 30 40 50 60 70

Godunov HLLEMCC

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HLLEM HLLE

Sod Problem: Contact Discontinuity

0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42

0.6 0.65 0.7 0.75 0.8 0.85

HLLEMCC HLLEM HLLE

Conclusions

• No complete analysis available

• Possible to avoid carbuncle while retaining exact resolution of contact waves

• No information on neighbouring Riemann problems needed (efficiency)

• Steady profiles replaced by stable neighbouring profiles

Appendix

Comparison of Roe and HLLE Flux

Roe Flux

g_Roe(q_r, q_l) = 1

2[f(q_r) + f(q_l)] − 1

2|A(q˜ _r, q_l)|(q_r − q_l)

HLLE Flux

g_HLL(q_r, q_l) = 1

2[f(q_r) + f(q_l)] − 1 2

S_R + S_L

S_R − S_L[f(q_r) − f(q_l)] + S_RS_L

S_R − S_L(q_r − q_l)

If Roe Matrix exists

g_HLL(q_r, q_l) = 1

2[f(q_r) + f(q_l)] − 1 2

S_R + S_L S_R − S_L

A(q˜ _r, q_l)(q_r − q_l) + S_RS_L

S_R − S_L(q_r − q_l)

Comparison of Viscosity Matrices

Roe

V_Roe = |A(q˜ _r, q_l)|

HLLE

V_HLL = S_R + S_L S_R − S_L

A(q˜ _r, q_l) − 2 S_RS_L S_R − S_LI

If S_L eigenvalue of A˜ with eigenvector r_l

V_HLLr˜_L = −S_Lr˜_L

If S_R eigenvalue of A˜ with eigenvector r_r

V_HLLr˜_r = S_Rr˜_r

Idea of HLLEM: Write Roe as Correction of HLL

• Choose S_L, S_R to be the Roe eigenvalues for the outer waves

• L˜ and R˜ Matrices with left/right eigenvectors of Roe matrix as rows/columns and L˜R˜ = I

• Find diagonal matrix K such that

V_Roe = V_HLL + S_RS_L S_R − S_L

RK˜ L˜

• Only entries for inner waves in K