high order finite volume schemescroisil/talk/fh5_26jan2011.pdfhigh order finite volume schemes...

HIGH ORDER FINITE VOLUME SCHEMES

Jean-Pierre Croisille

Laboratoire de mathematiques, UMR CNRS 71 22Univ. Paul Verlaine-Metz

LMD, Jan. 26, 2011

Jean-Pierre CROISILLE - Labo. Math., UMR 7122, Univ. Metz, F rance High order finite-volume schemes

Joint work withB. Courbet, F. Haider, ONERA, DSNA (Numerical Simulation of FluidFlows: Aerodynamics & Aeroacoustics, Chatillon-sous-Bagneux,France


Scientific context

High order numerical methods in Computational Fluid Dynamics

Today, finite volume methods are commonly used in many areas of CFD:aerodynamics (external flows) , aerothermochemistry(internal flows), porousmedia flows (petroleum industry, water resource), magnetohydrodynamics(cosmology).

Main reasons: versatility of the method, good properties (treatment ofconvective/diffusive flows, nonlinear waves, conservativity,) strong support in themathematical applied community.

Strong support in physics AND mathematics.

Strong interest for this kind of approach in the community in climatology. Generalcontext: development of the “dynamical core” of the GCM equations. Main model:the SW equations on the rotating spherical earth, 2D or 3D.

Since several years: strong interest for this problem in the mathematical/CFDcommunity.


Outline

1 A finite volume code - The package CEDRE (ONERA- DSNA):aerothermochemistry and energetics.

2 The MUSCL scheme for conservation laws

3 High order finite volumes methods

4 Perspectives: The MUSCL scheme on a spherical grid


The package CEDRE: Aerothermochemistry forcomplex fluid flows

Problems to solve

Full 3D Navier-Stokes equations + chemical reactions + turbulence modeling (RANS orLES) in complex geometries.

8

<

:

ρt + ∇ · (ρv) = 0

(ρv)t + ∇ · (ρv × v) + ∇p−∇ · τ) = 0

(ρetot)t + ∇ · (ρvetot + pv − τ · v + qth) = 0

Fluid models

Pressure law: p = p(ρ, ε).

Viscous strain tensor

τ(x, t) = λL∇ · v(x, t)δ(2) + 2µL∇⊙ v(x, t) (1)

λL, µL are the Lame coefficients, ⊙ denotes the symmetric tensor product, δ(2)

is the unit tensor.

Large scale turbulence modeling: LES or RANS.

Many other models: chemistry, diphasic flows, particles (eulerian or lagrangian),advected passive scalars.


Main features of CEDRE

Spatial approximation

General polyhedral grid

Cell-centered FV method: one unknown by cell.

Multidomain with parallelism

Many pysical models available: single phase flows, multiphasic flows, real gasflows, chemistry, particles (eulerian or lagrangian).


Typical set of conservative variables

Conservative variables “physical variables”

q =

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

ρ1

· · ·

ρnesp

−−−−−−−−

ρv1

ρv2

ρv3

−−−−−−−−−

ρet

−−−−−−−−

ρz1· · ·

ρznsca

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

u =

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

p

T

y1

· · ·

ynesp

−−−−−−−−

v1

v2

v3

−−−−−−−−−

ρz1· · ·

ρznsca

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

=

2

6

6

6

6

6

6

6

6

6

6

6

4

thermo. state−−−−−−−−

Velocity−−−−−−−−

Scalar values:Passive scalars

−−−−−−−−

RANS scalars

3

7

7

7

7

7

7

7

7

7

7

7

5

(2)


Spatial approximation: average on the cells

Semi-discrete scheme

Form of the semi-discrete dynamical system

˙Viqi = −X

j∈V

Ai,jfn,i→j −X

j∈V

Aijϕn,i→j + Viσi (3)

Upwinded Euler flux

fn,i→j = fn(ulimKi

, ulimKj

) =

8

<

:

RoeFlux “Low Mach” (Turkel)AUSM

(4)

Navier-Stokes diffusive flux ϕn,i→j = ϕn(uK ,∇uK)

Sources σi = σi(ui,∇ui).


Specificities in CEDRE (B. Courbet,DSNA-ONERA)

Specificities

1 Notion of “boundary cells” supporting a differential equation. Plays the same role

than an internal cell.

Useful for multidomains.Useful for boundary conditions. The boundary conditions using relaxationdifferential equations supported in boundary cells.

2 A geometric domain is “multisolver”. Coupling of different systems of equationsinside a single geometric domain.


Time algorithms

Explicit time-schemes

Runge-Kutta time schemes. Order 2,3,4.

Implicit time-schemes

Implicit linearized Euler scheme to compute asymptotic stationary states.

Linear solvers: GMRES with digonal preconditionning. Implicit time-dependent

schemes

2 or 3 implicit steps.Use of the approximate Jacobian operator based on the first order spatialscheme.Low Mach number flows: CFLhydro = 1 for the advective waves (shocks),CFLacc = 10 − 100 for the acoustic waves.


Main steps of the algorithm

Basic scheme

The scheme reads

Un+1i = Un

i −∆t

|Ti|

X

ij

|Aij |F (Uni , U

nj ) (5)

Given the values qi in the cells, interpolate the gradient ∇qi in each cells.

Compute the value of the piecewise linear function

Unj (x) = Un

j + ∇Unj (x− xj) (6)

Evaluate the numerical flux-function F (Uni , U

nj ). (Slope limiters are used).

Assemble the contribution of each interface Aij to the cells Ti and the cell Tj .

Update Uni by Un+1

i .

Additional features

The computing algorithm is much more complicated than expected at first glance! Loop on the faces, indirection problems, efficiency problems, interdomainproblems.

Accuracy of quadrature formulas on cell interfaces.


Simulations in aerothermochemistry with CEDRE


Local geometry of a cell in CEDRE


The MUSCL method in space

Conservation law

∂tu (x, t) + ∇ · f (u (x, t)) = 0 (7)

Local geometry

The cell with number α is denoted Tα, with barycenter xα and d-volume |Tα| . Theface Aαβ , with barycenter xαβ , has a normal vector nαβ oriented from cell Tα to Tβ

and of length‚

‚nαβ

‚

‚ equal to the surface˛

˛Aαβ

˛

˛. The oriented normal unit vector ofthe face Aαβ is ναβ .

Geometric notational convention

The following convention simplifies the notation of sums over cells. Whenever two cellshave no common interface, nαβ = 0 and kαβ = 0 and the face Aαβ is defined to beempty so that any surface integral over Aαβ is automatically zero. In addition, nαα,kαα and hαα are defined to be zero.


The MUSCL method in space, (cont.)

MUSCL with Method Of Lines

The simplest finite-volume scheme consists in evolving the quantities uα (t)

approximating the exact averages uα (t) with the dynamical system

duα (t)

dt= −

1

|Tα|

X

β

Z

Aαβ

fαβ

`

uα (t) , uβ (t)´

dσ (8)

MUSCL dynamical system

The numerical flux function fαβ (wint, wext) depends on the two states wint and wext oneach side of the cell interface. Conservation is translates as

fαβ (wint, wext) = −fβα (wext, wint) (9)


The MUSCL method in space, (cont.)

Final form of the MUSCL scheme

duα (t)

dt= (10)

−1

|Tα|

X

β

X

q

ωq fαβ

`

wα [u(t)]`

xαβ;q

´

, wβ [u(t)]`

xαβ;q

´´

.

Accuracy

In equation (10), the xαβ;q are the quadrature points on Aαβ and the ωq are thequadrature weights. If the expression under the integral on the right hand side of (8) isa polynomial of degree p in x and if the quadrature formula integrates exactly suchpolynomials, this step does not introduce any new discretization errors.


Cell-centered gradient reconstruction

Interpolation theory: cell-centered gradient reconstruction on general grids

A general theory of the gradient reconstruction is available (F. Haider). This theory isused in practice in CEDRE.

Consistency of the slope

Suppose given a function u(x) given by the (exact) averages uα on a given grid madeof VF cells Tα. Suppose given a linear “slope reconstruction” operator

u 7→ σα [u] =X

β

sαβ

`

uβ − uα

´

. (11)

The first order consistency is equivalent to the tensor equation

σ =X

β

sαβ

`

hαβ · σ´

for all σ ∈ Rd . (12)

or equivalentlyX

β

sαβ ⊗ hαβ = Id×d . (13)


Least-square gradient reconstruction

Proposition

Let the matrix Hα have rank d and let σα ∈ Rd be the solution of the least-squaresproblem

minσ∈Rd

8

<

:

X

β∈Wα

`

uβ − uα − hαβ · σ´2

9

=

;

. (14)

Then σα is unique and given by coefficients sαβ that are the columns of the minimumFrobenius norm solution to equation

SαHα = Id×d . (15)


Linear spectral analysis

Asymptotic stability

The stability property relevant for practical applications keeps to be the “linearasymptotic stability”. The semi-discrete MUSCL discrete form of the linear advectionequation

∂tu (x, t) + c · ∇u (x, t) = 0 , (x, t) ∈ Rd × R+ . (16)

gives a dynamical system

duα (t)

dt=X

β

Jαβuβ (t) ; 1 ≤ α ≤ N. (17)


Stability of the MUSCL scheme

Matrix of the MUSCL scheme

The operator J of the MUSCL scheme (17) is

Jαβ = − |Tα|−1

8

<

:

X

γ

(c · nαγ)+ δαβ +`

c · nαβ

´

−(18)

+X

γ

(nαγ · c)+ kαγ · sαβ −X

γ

(nαγ · c)+ kαγ · sα δαβ

−X

γ

(nγα · c)+ kγα · sγβ +`

nβα · c´

+kβα · sβ

9

=

;

.


Stability of the MUSCL scheme, (cont.)

Proposition (1)

[ Stability of Linear Systems]The system (17) is stable in the sense that

C = supt≥0

‖exp (tJ)‖ < ∞ (19)

if and only if all eigenvalues λ of J satisfy

1 Re (λ) ≤ 0 where Re (λ) is the real part of λ.

2 if Re (λ) = 0 then the Jordan index ı (λ) = 1 where ı (λ) is the maximaldimension of the Jordan blocks of J containing λ.

Definition ( Stable finite-volume operator )

The spatial discretization operator J in (18) are called stable if all their eigenvaluessatisfy properties (i)-(ii) of Proposition (1) for all advection velocities c ∈ Rd.


Localization of the eigenvalues of the MUSCL scheme

Problem

As usual in physics, we must solve the question of the localization of the eigenvalues ofa linear problem. Here the matrix to analyze contains:

The physics of the hyperbolic equation to solve

The upwinding (numerical flux)

The reconstruction operator

The most difficult: the geometry of the grid. The shape of the grid influences theshape of the spectrum !


Localization of the eigenvalues of the MUSCL scheme(cont.)

Classical tools

Fourier analysis. Not possible on irregular grids.

Direct algebraic eigenvalues analysis. Gerschgorin discs, field of values analysis:too weak.

Energy methods, Lyapunov theorems.


Extended Lyapunov Theorem

Theorem ( Extended Lyapunov Theorem)

Let J ∈ MN (C). The following properties are equivalent

1 J satisfies the conditions of Proposition 0.2, that is all eigenvalues λ of J haveRe (λ) ≤ 0 and if Re (λ) = 0 then the Jordan index ı (λ) = 1.

2 There exists a positive definite matrix G such that the matrix Q = GJ + J∗G isnegative semidefinite.



Corollaire

Consider the initial value problem

du(t)

dt= Ju(t) , u(0) = u0 , u(t) ∈ C

N , J ∈ MN (C) . (20)

Then there is a constant C such that

‖u(t)‖ ≤ C ‖u0‖ for all t ≥ 0

if and only if there exists a positive definite matrix G such that the matrixQ = GJ + J∗G is negative semidefinite.

Practical spectral analysis

It turns out to be very difficult to find a matrix G that could give a rigorous proof ofeigenvalue stability for the MUSCL operator J .



Table: Least-Squares Reconstruction on the First Neighborhood in 3D:spectral abscissa ωJ and statistics of ‖Rα‖L(2,∞)

grid spectral abscissa average maximum 90th percentile

tetrahedral 1 1.6539 0.57376 0.99051 0.67154tetrahedral 2 -0.46968e-10 0.57143 1.0878 0.67217tetrahedral 3 5.7716 0.56804 1.0533 0.65979tetrahedral 4 7.5288 0.57435 1.0888 0.67144

hybrid 1 2.1612 0.54796 1.0820 0.65732hybrid 2 5.5859 0.55320 1.0702 0.68159hybrid 3 6.5645 0.53307 1.0962 0.66178hybrid 4 7.2591 0.52921 1.1547 0.64271

deformed cartesian 1 -0.17017e-9 0.40784 0.54825 0.45403deformed cartesian 2 0.42669e-10 0.41018 0.54821 0.45641deformed cartesian 3 -0.63580e-10 0.41188 0.58191 0.46029deformed cartesian 4 -0.55940e-10 0.41334 0.56309 0.46198


Numerical experiment

Figure: Tetrahedral grid . Unstable spectrum. Convection equation.


Numerical experiment

Figure: Tetrahedral grid of a channel. The flow is along direction x.


Numerical experiment (cont.)

0 1 2 3 4 5 6 7 8−10

−8

−6

−4

−2

0

2

4

6

8

10

time

log

10

(re

sid

ua

ls)

ρρ v

x

ρ vy

ρ vz

ρ E

Figure: Residuals history of ddt

[ρ, ρux, ρuy, ρuz, ρE], 800 time iterations, firstneighborhood interpolation of the gradient, no slope limiters.


Numerical experiment (cont.)

0 5 10 15 20 25 30 35 40−10

−8

−6

−4

−2

0

2

4

6

8

time

log

10

(re

sid

ua

ls)

ρρ v

x

ρ vy

ρ vz

ρ E

Figure: Residuals history of ddt

[ρ, ρux, ρuy, ρuz, ρE], 4000 time iterations,second neighborhood interpolation of the gradient, no slope limiters.


High order finite volume MUSCL schemes

Observation

The MUSCL scheme with piecewise linear reconstruction is “second order”. Thisis sufficient for accurate computation of non linear waves such as shocks.

This is dramatically insufficient for the computation of the fundamentalcharacteritics of diffusive turbulent flows.

The conclusion is that we need to have higher order spatial approximations.

Question

Is it possible to switch from a piecewise linear reconstruction to a higher orderreconstruction within the MUSCL framework?

Or should we give up the MUSCL approach (one unknown per cell) and switch toalternative “high order” schemes like the spectral element (Patera, Deville), thespectral difference (Wang,Liu,Vinokur) or the Discontinuous Galerkin schemes(Cockburn, Shu),... which all use more than one unknown per cell.

The question to build higher order MUSCL schemes on irregular grids wasalready studied in the 90’ (Harten, Osher, Abgrall,...).

This keeps to be a question of fundamental practical importance !


High order MUSCL method

Interpolation on irregular grids

Basic methodology: interpolate the first, second and third order derivatives in acell.

The mathematical problem belongs therefore to the interpolation theory onunstructured grids.

General theory of conservative interpolation on unstructured finite volume gridscan be found in the thesis of F. Haider (2009, Univ. Paris 6).

Fourth order MUSCL schemes

The reconstructed function ui(x) is

ui(x) = ui + σi(x− xi) +1

2θi(x− xi)

2 +1

6ψi(x− xi)

3 (21)

We haveσi ≃ u′(xi), θi ≃ u′′(xi), ψi ≃ u′′′(xi) (22)

The scheme is as before:

Un+1i = Un

i −∆t

|Ti|

X

ij

|Aij |F (Uni , U

nj ) (23)


Computation of the reconstruction

Least square approximation

The least-square slope is8

>

<

>

:

σLSα =

P

β∈Vαcαβ(uβ − uα), (a)

θLSα =

P

β∈Vαcαβ(σβ − σα), (b)

ψLSα =

P

β∈Vαcαβ(θβ − θα), (c)

(24)

where the coefficients aα, bα and aα depend only of the local shape of the grid.

Approximate derivatives

Suppose that σα = u′(xα), θα = u′′(xα), ψα = u′′′(xα), Then approximations ofσα,θα, ψα of order 3, 2, 1 are given by the explicit formulas are given by8

>

>

>

<

>

>

>

:

ψα =P

β c∗αβ

P

γ∈Vβcβγ

`

σLSγ − σLS

β

´

−P

δ∈Vαcαδ(σLS

δ − σLSα )

!

, (a)

θα = −aαψα +P

β cαβ(σLSβ − σLS

α ), (b)

σα = −aαθα − bαψα + σLSα , (c)

(25)


Computational algorithm

ALGORITHM 1

1 Compute σLSα in all Tα .This involves the stencil Vα only.

2 ComputeP

β∈Vαcαβ(σLS

β− σLS

α ) in all cells Tα.

3 Compute the third order derivative ψα by (25)(a).

4 Compute the second order derivative θα by (25)(b).

5 Compute the first order derivative σα by (25)(c).

Order of the interpolating polynomial

The polynomial

pα(x) = uα + σα(x− xα) + θα

1

2(x− xα)2 −

1

24|Tα|

2

!

+1

6ψα(x− xα)3 (26)

is a fourth order reconstruction of u(x) in the cell Tα.


Full discrete fourth order MUSCL scheme

4th order scheme

Use the ordinary RK4 scheme.8

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

:

k0 = JUn

k1 = J(Un + 12∆t k0)

k2 = J(Un + 12∆t k1)

k3 = J(Un + ∆t k2)

Un+1 = Un + ∆t

16k0 + 1

3k1 + 1

3k2 + 1

6k3

!

(27)

Observation: this scheme is stable on highly irregular random one-dimensionalgrid with random size.

On a smooth varying grid the measured convergence rates are shown on thefollowing table


Assessing the fourth order accuracy of the MUSCLscheme

Rate of convergence of the 4th order MUSCL scheme - Sinusoidal grid.

Period N=32 rate N=64 rate N=128 rate N=256 rate N=512

1 2.62(-4) 3.85 1.81(-5) 3.68 1.41(-6) 3.85 9.72(-8) 3.95 6.26(-9)5 1.17(-3) 3.82 8.28(-5) 3.56 6.99(-6) 3.84 4.86(-7) 3.95 3.13(-8)10 2.33(-3) 3.82 1.64(-4) 3.56 1.39(-5) 3.83 9.75(-7) 3.95 6.26(-8)


The cubed sphere grid

The cubed sphere grid

Spherical grid made of 6 squares patches. The 6 patches are the projection of acube on an (internal) sphere. Nonequiangular grid.

Very interesting to discretize the SW system.

First attempt recently done (Ulrich, Jablonowski, van Leer, JCP, 2010). Goodresults with some 4th order MUSCL scheme. Classical test-case (Williamson) aretested.


The cubed sphere grid (cont.)

−1

−0.5

0

0.5

1

−1

−0.5

0

0.5

1−1

−0.5

0

0.5

1

F

W

S

N

E

B

−1

0

1

−1−0.500.51

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

W

S

F

N

B

E

Figure: Cubed sphere grids, C8 and C32


Implementation so far

CEDRE

Second order MUSCL scheme in CEDRE (aerothermochemistry for internalflows).

Various options of turbulence modling

Various options of time-stepping

Irregular general grids

Multisolver/multidomain solver

Fourth order accuracy

Methodology of higher order accurate MUSCL scheme assessed on tetrahedralgrids (F. Haider).

Keeps the standard MUSCL framework,one unknown per cell.

Extension to flows on the spherical earth

Methodology seems possible on any kind of spherical grids: latitude/longitude,web grid, icosahedral grid, etc.

A specifically interesting grid seems to be the “cubed sphere grid”.


In progress

At ONERA

Implementation of the fourth order MUSCL scheme for 3D gas dynamics problems(ONERA, F. Haider, B. Courbet).

Cubed sphere grid

High order Hermitian interpolation on the cubed sphere grid;

Fast solver (FFT)


high order finite volume schemescroisil/talk/fh5_26jan2011.pdfhigh order finite volume schemes...

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