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Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation Laws Andreas Meister UMBC, Department of Mathematics and Statistics Andreas Meister (UMBC) Finite Volume Scheme 1/1
56

Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Jun 22, 2020

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Page 1: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Implicit Finite Volume Schemes andPreconditioned Krylov Subspace Methods

for the Discretization ofHyperbolic and Parabolic

Conservation Laws

Andreas Meister

UMBC, Department of Mathematics and Statistics

Andreas Meister (UMBC) Finite Volume Scheme 1 / 1

Page 2: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Content

Aims

Navier Stokes Equations

Finite Volumen Scheme

Iterative Solution Methods and Preconditioning

Numerical Results

Andreas Meister (UMBC) Finite Volume Scheme 2 / 1

Page 3: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Aims

Development of a numerical method for the simulation of reallife fluid dynamics problems in the project CASWING

Flexible usage as a basic method

- Turbulent combustion

- Chemical reacting fluid flows

- Low Mach number fluid flows

- Thermoregulation

- Fluid Structur Interaction

Investigation of methods to accelerate the convergence

Andreas Meister (UMBC) Finite Volume Scheme 3 / 1

Page 4: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Aims

Requirements

- Flow dependent discretization of arbitrary complex geometries

- Taking account of moving boundaries

- Consideration of turbulence effects

- Minimization of computing time? Implicit scheme

? Preconditioned Krylov subspace methods

Andreas Meister (UMBC) Finite Volume Scheme 4 / 1

Page 5: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Balance laws

Unsteady, compressible and dimesionless Navier Stokes equations

∂tu +2∑

m=1

∂xmgcm(u) =

1Re∞

2∑m=1

∂xm fνm(u)

Vector of conserved variables

u = (ρ, ρv1, ρv2, ρE)

Convective flux function

gcm(u) =

ρvm

ρvmv1 + δm1 p

ρvmv2 + δm2 p

ρHvm

Viscous flux function

fνm(u) =

0

µS1mµS2m∑3

i=1 µSimvi + µκPr∞∂xm e

m = 1, 2

Andreas Meister (UMBC) Finite Volume Scheme 5 / 1

Page 6: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Finite Volume Method∫σ(t)

∂tu dx +2∑

m=1

∫σ(t)

∂xmgcm(u) dx =

1Re∞

2∑m=1

∫σ(t)

∂xm fνm(u) dx

Cell averages on a dual mesh

ui =1|σi |

∫σi

u dx

Gauß Integral Theorem and Reynolds Transport Theorem(Evolution equation for cell averages)

ddt

ui =1|σi(t)|

1Re∞

∫∂σi (t)

2∑m=1

fνm(u)nm ds −∫

∂σi (t)

2∑m=1

fcm(u)nm ds

fcm(u) = gc

m(u)− νnetz,mu

Andreas Meister (UMBC) Finite Volume Scheme 6 / 1

Page 7: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Discretization of the convective fluxes

ddt

ui =1|σi |

1Re

∫∂σi

2∑m=1

fνm(u)nmds −∫∂σi

2∑m=1

fcm(u)nmds

Representation of the boundary integrals∫∂σi

2∑m=1

fcm(u)nmds =

∑j∈N(i)

2∑k=1

∫lkij

2∑m=1

fcm(u)nk

ij,m ds

Quadrature rule∫lkij

2∑m=1

fcm(u)nk

ij,m ds ≈ Hc(ui ,uj ; nkij )|lkij |

Convective flux function Hc (AUSMDV, Lax-Friedrichs, ...)Spiatial discretization of higher order

TVD-scheme, WENO- und ENO-method

Andreas Meister (UMBC) Finite Volume Scheme 7 / 1

Page 8: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Discretization of the convective fluxes

ddt

ui =1|σi |

1Re

∫∂σi

2∑m=1

fνm(u)nmds −∫∂σi

2∑m=1

fcm(u)nmds

Representation of the boundary integrals∫∂σi

2∑m=1

fcm(u)nmds =

∑j∈N(i)

2∑k=1

∫lkij

2∑m=1

fcm(u)nk

ij,m ds

Quadrature rule∫lkij

2∑m=1

fcm(u)nk

ij,m ds ≈ Hc(ui ,uj ; nkij )|lkij |

Convective flux function Hc (AUSMDV, Lax-Friedrichs, ...)Spiatial discretization of higher order

TVD-scheme, WENO- und ENO-method

Andreas Meister (UMBC) Finite Volume Scheme 7 / 1

Page 9: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Discretization of the convective fluxes

ddt

ui =1|σi |

1Re

∫∂σi

2∑m=1

fνm(u)nmds −∫∂σi

2∑m=1

fcm(u)nmds

Representation of the boundary integrals∫∂σi

2∑m=1

fcm(u)nmds =

∑j∈N(i)

2∑k=1

∫lkij

2∑m=1

fcm(u)nk

ij,m ds

Quadrature rule∫lkij

2∑m=1

fcm(u)nk

ij,m ds ≈ Hc(ui ,uj ; nkij )|lkij |

Convective flux function Hc (AUSMDV, Lax-Friedrichs, ...)Spiatial discretization of higher order

TVD-scheme, WENO- und ENO-method

Andreas Meister (UMBC) Finite Volume Scheme 7 / 1

Page 10: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Discretization of the convective fluxes

ddt

ui =1|σi |

1Re

∫∂σi

2∑m=1

fνm(u)nmds −∫∂σi

2∑m=1

fcm(u)nmds

Representation of the boundary integrals∫∂σi

2∑m=1

fcm(u)nmds =

∑j∈N(i)

2∑k=1

∫lkij

2∑m=1

fcm(u)nk

ij,m ds

Quadrature rule∫lkij

2∑m=1

fcm(u)nk

ij,m ds ≈ Hc(ui ,uj ; nkij )|lkij |

Convective flux function Hc (AUSMDV, Lax-Friedrichs, ...)Spiatial discretization of higher order

TVD-scheme, WENO- und ENO-method

Andreas Meister (UMBC) Finite Volume Scheme 7 / 1

Page 11: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Discretization of the viscous fluxes

ddt

ui =1|σi |

1Re

∫∂σi

2∑m=1

fνm(u)nmds −∫∂σi

2∑m=1

fcm(u)nmds

fνm(u) =

0τ1mτ2m

v1τ1m + v2τ2m + κγ(γ−1)Pr ∂xmT

Values at the midpoint xk

ij of the line segment lkij :

Velocity and viscosityGradient of the velocity and the temperature

1Re

∫lkij

2∑m=1

fνm(u)nm ds ≈2∑

m=1

Hνm (v, µ,∇xv,∇xT ) nm |lkij |

Andreas Meister (UMBC) Finite Volume Scheme 8 / 1

Page 12: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

ui =1|σi |

1Re

∫∂σi

2∑m=1

fνm(u)nmds −∫∂σi

2∑m=1

fcm(u)nmds

Numerical methods for ordinary differential equations

Runge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditionedAndreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 13: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

ui =1|σi |

∑j∈N(i)

2∑k=1

1Re

∫lkij

2∑m=1

fνm(u)nmds −∫lkij

2∑m=1

fcm(u)nmds

Numerical methods for ordinary differential equations

Runge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditionedAndreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 14: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

ui =1|σi |

∑j∈N(i)

2∑k=1

2∑m=1

Hνm (u) nm |lkij | −

∫lkij

2∑m=1

fcm(u)nmds

Numerical methods for ordinary differential equations

Runge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditionedAndreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 15: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

ui =1|σi |

∑j∈N(i)

2∑k=1

(2∑

m=1

Hνm (u) nm |lkij | − Hc(ui ,uj ; nk

ij )|lkij |

)

Numerical methods for ordinary differential equationsRunge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditionedAndreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 16: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

ui = Fi (u) , i = 1, . . . ,N

Numerical methods for ordinary differential equationsRunge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditioned

Andreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 17: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

u = F (u) , u ∈ RRR4∗N

Numerical methods for ordinary differential equationsRunge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditioned

Andreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 18: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

u = F (u) , u ∈ RRR4∗N

Numerical methods for ordinary differential equationsRunge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditioned

Andreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 19: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

u = F (u) , u ∈ RRR4∗N

Numerical methods for ordinary differential equationsRunge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditioned

Andreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 20: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

u = F (u) , u ∈ RRR4∗N

Numerical methods for ordinary differential equationsRunge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditioned

Andreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 21: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

u = F (u) , u ∈ RRR4∗N

Numerical methods for ordinary differential equationsRunge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditioned

Andreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 22: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

u = F (u) , u ∈ RRR4∗N

Numerical methods for ordinary differential equationsRunge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditioned

Andreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 23: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Time stepping scheme

ddt

u = F (u) , u ∈ RRR4∗N

Numerical methods for ordinary differential equationsRunge-Kutta-methods (Euler-scheme, DIRK, SDIRK)Adams-, Nyström-, Milne-Simpson-schemeBDF-method

Explicit ansatz un+1 = un + ∆t F (un)

Easy to implementRestrictive CFL-condition

Implicit ansatz un+1 = un + ∆t F(un+1)

Large time step size ∆tSolution of a (non-)linear system

→ Large, sparse, badly conditioned

Andreas Meister (UMBC) Finite Volume Scheme 9 / 1

Page 24: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Iterative solution methods

Consider Ay = b with A ∈ RRRn×n nonsingular , y ,b ∈ RRRn

Direct methods

- Gauß-elmination (LU-factorization)Cholesky-factorization (LLT -factorization)QR-factorization

- Why do we calculate the exact solution?

- Rounding errors, Storage requirements

Andreas Meister (UMBC) Finite Volume Scheme 10 / 1

Page 25: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Iterative solution methods

Consider Ay = b with A ∈ RRRn×n nonsingular , y ,b ∈ RRRn

Iterative methods

- Splitting-methods? Jacobi-, Gauß-Seidel-, SOR-schemes

? Easy to implement, slow convergence

- Krylov subspace schemes? CG, GMRES, CGS, Bi-CGSTAB, TFQMR, QMRCGSTAB

? Higher stability and often faster convergence

- Condition number? Equivalent transformation of the linear system

(Preconditioning)

Andreas Meister (UMBC) Finite Volume Scheme 11 / 1

Page 26: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Preconditioning

Equivalent transformation of the linear system

Ay = b

inPlAPr z = Plb

y = Pr z

to accelerate the convergence.Left preconditioning: Pl 6= IRight preconditioning: Pr 6= IBoth sided preconditioning: Pr 6= I 6= Pl

Control of the residual:

Residual right preconditioning: r rm = b − APr zm = b − Aym = rm

Residual left preconditioning: r lm = Plb − PlAzm = Pl rm

Andreas Meister (UMBC) Finite Volume Scheme 12 / 1

Page 27: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Preconditioning

Incomplete LU-factorizationAdditional storage, time consuming calculationFrozen formulation, here ILU(p)

Splitting schemes (GS, SGS)No additional storageNo calculation

Scaling (Jacobi et. al.)Low additional storage, weak acceleration

Characteristic approachEigenvalues λ1/2 = v · n, λ3/4 = v · n + cRenumberingApplication of a splitting-based preconditionerCHASGS and CHAGS

Andreas Meister (UMBC) Finite Volume Scheme 13 / 1

Page 28: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Bi-NACA0012 Airfoil

Ma = 0.55, Angle of attack 6, inviscidTriangulation: 26632 triangles, 13577 points

Abbildung: Partial view of the triangulation w.r.t. the Bi-NACA0012 Airfoil anddensity distribution

Andreas Meister (UMBC) Finite Volume Scheme 14 / 1

Page 29: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Bi-NACA0012 Airfoil

Ma = 0.55, Angle of attack 6, inviscidTriangulation: 26632 triangles, 13577 points

Abbildung: Distribution of the Mach number at the surface of the upper (left)and lower (right) airfoil

Andreas Meister (UMBC) Finite Volume Scheme 15 / 1

Page 30: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Combustion chamber at supersonic speed

Ma = 2.0, Angle of attack 0, inviscidTriangulation: 61731 triangles, 31202 points

Abbildung: Adaptive triangulation and isolines of the density

Abbildung: Distribution of the Mach number and the pressure

Andreas Meister (UMBC) Finite Volume Scheme 16 / 1

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NACA0012-Airfoil

Ma = 0.85, α = 0, viscousRe = 500, T∞ = 273K , adiabatic

Abbildung: Partial view of the triangulation w.r.t. the NACA0012 Airfoil andisolines of the Mach number

Andreas Meister (UMBC) Finite Volume Scheme 17 / 1

Page 32: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

NACA0012-Airfoil

Cp-distribution

Abbildung: Cp-distribution unstructure implicit DLR- TAU-Code and structuredexplicit ViB-Code

Andreas Meister (UMBC) Finite Volume Scheme 18 / 1

Page 33: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Shock-Shock-Interaction

Shock-Shock-InteractionMa = 8.03, Re = 193.750, T∞ = 122.1K

-100.0 -50.0 0.0 50.0 100.0Ψ[deg]

0.0

2.0

4.0

6.0

8.0

10.0

12.0

p/p0

τ-Code Experiment

-100.0 -50.0 0.0 50.0 100.0Ψ[deg]

0.0

2.0

4.0

6.0

8.0

10.0

q/q0

τ-CodeExperiment

Adaptive triangulation and isolines of the Mach numberdistribution of the heat flux and the pressure

Andreas Meister (UMBC) Finite Volume Scheme 19 / 1

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Shock-Shock-Interaction

Stoß-Stoß-WechselwirkungMa = 8.03, Re = 193.750, T∞ = 122.1K

Adaptive triangulation, isolines of the Mach numberand temperature distribution

Andreas Meister (UMBC) Finite Volume Scheme 20 / 1

Page 35: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

RAE 2822 Airfoil

Ma = 0.75, Angle of attack 3, inviscidTriangulation: 9974 triangles, 5071 points

Abbildung: Distribution of the density and Cp-distribution

Andreas Meister (UMBC) Finite Volume Scheme 21 / 1

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RAE 2822 Airfoil

Ma = 0.75, Angle of attack 3, inviscidTriangulation: 9974 triangles, 5071 points

Explicit scheme Implicit scheme

Scaling Incomplete LU(5)

100 % 24,36 % 2,86 %

Tabelle: Percentage comparison of the computing time

Andreas Meister (UMBC) Finite Volume Scheme 22 / 1

Page 37: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

SKF1.1 Airfoil

Ma = 0.65, Angle of attack 3, inviscidTriangulation: 46914 triangles, 23751 points

Abbildung: Distribution of the density and Cp-distribution

Andreas Meister (UMBC) Finite Volume Scheme 23 / 1

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SKF1.1 Airfoil

Ma = 0.65, Angle of attack 3, inviscidTriangulation: 46914 triangles, 23751 points

Explicit scheme Implicit scheme

Scaling Incomplete LU(5)

100 % 68,58 % 9,97 %

Tabelle: Percentage comparison of the computing time

Andreas Meister (UMBC) Finite Volume Scheme 24 / 1

Page 39: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Convergence analysis

Convergence criterion

Residuum =14t

√√√√#Nh∑i=1

(4ρ)2i |σi ||Ω|

Convergence history using ILU-preconditioned schemes

Abbildung: Bi-NACA0012 Airfoil (left) and Combustion chamber (right)

Andreas Meister (UMBC) Finite Volume Scheme 25 / 1

Page 40: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Convergence analysis

Abbildung: NACA0012 Airfoil (left) and decay of homogeneous turbulence(right)

Andreas Meister (UMBC) Finite Volume Scheme 26 / 1

Page 41: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

Convergence analysis (ILU vs. Scaling)

Bi-NACA0012 Airfoil

NACA0012 Airfoil

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Convergence analysis (ILU vs. Scaling)

Laminar boundary layer of a flate plate: Re = 6 · 106, Ma = 5.0

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Preconditioned Bi-CGSTAB

Choose z0 ∈ RRRn and ε > 0r0 = p0 := Plb − PlAPr z0ρ0 := (r0, r0)2, j := 0While ‖rj‖2 > ε :

vj := PlAPr pj

αj :=ρj

(vj , r0)2sj := rj − αjvjtj := PlAPr sj

ωj :=(tj , sj )2(tj , tj )2

zj+1 := zj + αjpj + ωjsjrj+1 := sj − ωj tjρj+1 := (rj+1, r0)2

βj :=αj

ωj

ρj+1

ρjpj+1 := rj+1 + βj (pj − ωjvj )j := j + 1

yj = Pr zj

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Modified Bi-CGSTAB method

Choose y0 ∈ RRRn and ε > 0r0 = p0 := b − A y0 , rP

0 = pP0 := PL r0

ρP0 :=

(rP0 , r

P0

)2, j := 0

While ‖rj‖2 > ε :vj := A PR pP

j , vPj := PL vj

αPj :=

ρPj

(vPj ,r

P0 )2

sj := rj − αPj vj , sP

j := PLsj

tj := A PR sPj , tP

j := PL tj

ωPj :=

(tPj ,s

Pj )2

(tPj ,t

Pj )2

yj+1 := yj + αPj PR pP

j + ωPj PR sP

jrj+1 := sj − ωP

j tj ,rPj+1 := sP

j − ωPj tP

jρP

j+1 := (rPj+1, r

P0 )2,

βPj :=

αPj

ωPj

ρPj+1

ρPj

pPj+1 := rP

j+1 + βPj (pP

j − ωPj vP

j )

j := j + 1Andreas Meister (UMBC) Finite Volume Scheme 30 / 1

Page 45: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

SKF 1.1 two element airfoil

Ma = 0.65, α = 3.0

0.0 0.5 1.0−1.0

0.0

1.0

2.0

−Cp

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Page 46: Implicit Finite Volume Schemes and Preconditioned …Implicit Finite Volume Schemes and Preconditioned Krylov Subspace Methods for the Discretization of Hyperbolic and Parabolic Conservation

NACA0012 Airfoil

Ma = 0.85, α = 1.25

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Comparison of Preconditioners

SKF 1.1 two element airfoil; Ma = 0.65, α = 3.0

D1(r) D2(r) D∞(r) DJac(r) DJac(l)1038% 953% 992% 438% 514%

NACA0012 Airfoil Ma = 0.85, α = 1.25

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NACA0012 Airfoil

Ma = 3.0, α = 0

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Cylinder

Ma = 10.0, α = 0

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Comparison of Preconditioners

NACA0012 Airfoil Ma = 3.0, α = 0

Cylinder Ma = 10.0, α = 0

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Pitching Airfoil

Instationär – Inviscid

Airfoil : NACA0012

Freestream Mach number : 0.755

Angle of attack in rest : 0.016

Amplitude : 2.51

Frequency : 0.1628

Number of nodes : 7141

Number of triangles : 14005

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Pitching NACA0012 Airfoil, Density distribution

Ma = 0.755, α = 0.016 + 2.51sin kt , k = 0.1628, inviscid

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Pitching NACA0012 - Airfoil, Cp-distribution

Ma = 0.755, α = 0.016 + 2.51sin kt , k = 0.1628, inviscid

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Pitching NACA0012 - Airfoil, Cp-distribution

Ma = 0.755, α = 0.016 + 2.51sin kt , k = 0.1628, inviscid

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Pitching NACA0012 - Airfoil, Lift and Momentum

Lift coefficient Momentum coefficient

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Conclusion

Navier-Stokes Equations

Finite Volumen Scheme

AUSMDV-Riemannsolver for the convective fluxes

Central scheme for the viscous fluxes

Iterative system solver (BiCGSTAB)

Preconditioner (ILU(r,p), CHASGS(r))Results

- very good results for inviscid test cases for both stationary andmoving grids

- very good results for viscous test cases

Application of implicit time stepping schemes- Steady flow fields- Unsteady flow fields at low Mach number- Unsteady flow fields without fast moving shocks- Adaption CFL number

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