Vander Vegt

8/8/2019 Vander Vegt

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University of Twente - Chair Numerical Analysis and Computational Mechanics 1

Introduction

Motivation of research:

Many important flow phenomena are modelled by nonconservative (hyperbolic)partial differential equations, e.g. dispersed multiphase flows.

Our aim is to develop space-time discontinuous Galerkin discretizations which aresuitable for both conservative and nonconservative partial differential equations


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Introduction

Motivation of research:

Nonconservative hyperbolic partial differential equations contain nonconservativeproducts

x

u + A(u)x

u = 0

The essential feature of nonconservative products is that A = Df, hence A isnot the Jacobian matrix of a flux function f.

This causes problems once the solution becomes discontinuous, because the weaksolution in the classical sense of distributions then does not exist.

This also complicates the derivation of discontinuous Galerkin discretizations sincethere is no direct link with a Riemann problem.


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Alternative: use the theory for nonconservative products from Dal Maso, LeFlochand Murat (DLM)


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Overview of Presentation

Overview of main results of the theory of Dal Maso, LeFloch and Murat fornonconservative products

Space-time DG discretization of nonconservative hyperbolic partial differentialequations

Extension to the compressible Navier-Stokes equations

Multigrid techniques for space-time DG discretizations

Conclusions


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Nonconservative Products

Consider the function u(x)

u(x) = uL + H(x xd)(uR uL), x, xd ]a, b[,

with H : R

R the Heaviside function.

For any smooth function g : Rm Rm the product g(u)xu is not defined atx = xd since here |xu| .

Introduce a smooth regularization u of u. If the total variation of u remainsuniformly bounded with respect to then Dal Maso, LeFloch and Murat (DLM)

showed that

g(u)dudx lim

0 gududxgives a sense to the nonconservative product as a bounded measure.


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Effect of Path on Nonconservative Product

The limit of the regularized nonconservative product depends in general on the path

used in the regularization.

Introduce a Lipschitz continuous path : [0, 1] Rm, satisfying (0) = uLand (1) = uR, connecting uL and uR in R

m.

The following regularization u for u then emerges:

u(x) =

uL, if x ]a, xd [,(

xxd+2 ), if x ]xd , xd + [, > 0.

uR, if x ]xd + , b[


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When tends to zero, then:

g(u)du

dx Cxd, with C =

10

g(())d

d() d,

weakly in the sense of measures on ]a, b[, where xd is the Dirac measure at xd.

The limit of g(u)xu depends on the path .

There is one exception, namely if an q : Rm R exists with g = uq. In thiscase C = q(uR) q(uL).


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DLM Theory

Dal Maso, LeFloch and Murat provided a general theory for nonconservative

hyperbolic pdes.

Introduce the Lipschitz continuous maps : [0, 1]

R

m

R

m

R

m which

satisfy the following properties:

(H1) (0; uL, uR) = uL, (1; uL, uR) = uR,

(H2) (; uL, uL) = uL,

(H3)

(; uL, uR)

K|uL uR|, a.e. in [0, 1].


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Theorem (DLM). Let u :]a, b[ Rm be a function of bounded variation andg : Rm Rm a continuous function. Then, there exists a unique real-valuedbounded Borel measure on ]a, b[ with:

1. If u is continuous on a Borel set B ]a, b[, then

(B) = B g(u)du

dx

2. If u is discontinuous at a point xd of ]a, b[, then

({xd}) =

10

g((; uL, uR))

(; uL, uR) d.

By definition, this measure is the nonconservative product of g(u) by xu anddenoted by =

g(u)dudx

.


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Rankine-Hugoniot Relations

For conservative hyperbolic system of pdes, xu + xf(u) = 0 theRankine-Hugoniot relations across a jump with uL and uR and velocity v are

equal to

v(uR uL) + f(uR) f(uL) = 0.

For a nonconservative hyperbolic pde xu + A(u)xu = 0 the Rankine-Hugoniotrelations in the DLM theory are equal to

v(uR uL) +1

0

A(D(s, uL

, uR

))sD(s; uL

, uR

)ds = 0

with D a Lipschitz continuous path satisfying D(0; uL, uR) = uL andD(1; uL, uR) = uR.

The Rankine-Hugoniot relations are essential for the definition of the NCP fluxused in the DG discretization.


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Space-Time Discontinuous Galerkin Method

I nt n+1

tn

t0

Qn

n

n+1

y

x

t

K

A time-dependent problem is considered directly in four dimensional space, withtime as the fourth dimension


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Key Features of Space-Time Discontinuous Galerkin Methods

Simultaneous discretization in space and time: time is considered as a fourthdimension.

Discontinuous basis functions, both in space and time, with only a weak couplingacross element faces resulting in an extremely local, element based discretization.

The space-time DG method is closely related to the Arbitrary Lagrangian Eulerian(ALE) method.


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Benefits of Discontinuous Galerkin Methods

Due to the extremely local discretization DG methods provide optimal flexibility for

achieving higher order accuracy on unstructured meshes

hp-mesh adaptation

unstructured meshes containing different types of elements, such as tetrahedra,

hexahedra and prisms

parallel computing


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Benefits of Space-Time Discontinuous Galerkin Methods

A conservative discretization is obtained on moving and deforming meshes.

No data interpolation or extrapolation is necessary on dynamic meshes, at freeboundaries and after mesh adaptation.


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Disadvantages of Space-(Time) Discontinuous Galerkin Methods

Algorithms are generally rather complicated, in particular for elliptic and parabolicpartial differential equations

On structured meshes DG methods are computationally more expensive than finitedifference and finite volume methods.


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Space-Time Domain

Consider an open domain: E Rd. The flow domain (t) at time t is defined as:

(t) := {x E | x0 = t, t0 < t < T}

The space-time domain boundary E consists of the hypersurfaces:

(t0) :={x E | x0 = t0},(T) :={x E | x0 = T},

Q :={x E | t0 < x0 < T}.

The space-time domain is covered with a tessellation Th consisting of space-timeelements K.


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Discontinuous Finite Element Approximation

The finite element space associated with the tessellation Th is given by:

Wh :=

W (L2(Eh))m : W|K GK (Pk(K))m, K Th

The jump of f at an internal face S SnI in the direction k of a Cartesiancoordinate system is defined as:

[[f]]k = fLnLk + f

RnRk ,

with nRk = nLk .

The average of f at S Sn

I is defined as:

{{f}} = 12(fL

+ fR

).


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Space-Time DG Formulation of Nonconservative HyperbolicPDEs

Consider the nonlinear hyperbolic system of partial differential equations innonconservative form in multi-dimensions:

Ui

t +

Fik

xk + GikrUr

xk = 0, x Rq

, t > 0,

with U Rm, F Rm Rq, G Rm Rq Rm

These equations model for instance bubbly flows, granular flows, shallow waterequations and many other physical systems.


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Weak formulation for nonconservative hyperbolic system:

Find a U Vh, such that for V Vh the following relation is satisfiedKTh

K

Vi

Ui,0 + Fik,k + GikrUr,k

dK

+

KTh

K(tn+1)

Vi(URi ULi ) dK K(t+n )

Vi(URi ULi ) dK+

SSI

S

Vi 10

Gikr((; UL

, UR

))r

(; U

L, U

R) d n

Lk

dS

SSI

SVi[[Fik vkUi]]k dS = 0


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Relation with Space-Time DG Formulation of ConservativeHyperbolic PDEs

Theorem 2. If the numerical flux V for the test function V is defined as:

V = {{V}} at S SI,

0 at K(tn) h(tn) n 0,then the DG formulation will reduce to the conservative space-time DG formulation

when there exists a Q, such that Gikr = Qik/Ur.


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After the introduction of the numerical flux V we obtain the weak formulation:KTh

K

Vi,0Ui Vi,kFik + ViGikrUr,k dK+

KTh

K(tn+1)

VLi ULi dK

K(t+n )

VLi ULi dK

+

SSI

S

(VL

i VRi ){{Fik vkUi}}nLk dS

+

SSB

S

VLi (FL

ik vkULi )nLk dS

+ SSI

S{{Vi}} 1

0 Gikr((; U

L

, UR

))

r

(; UL

, UR

) d nLk dS = 0


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Numerical Fluxes

The fluxes at the element faces do not contain any stabilizing terms yet, both forthe conservative and nonconservative part

At the time faces, the numerical flux is selected such that causality in time isensured U = UL at K(tn+1)

UR at K(t+n ).

The space-time DG formulation is stabilized using the NCP (Non-ConservativeProduct) flux

Pnc

i = {{Fik vkUi}} + PiknLk


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Nonconservative Product Flux

E

T

dd

dd

dd

dd

dd

dd

ddd

xL xR

x

t

SL SRSM

T

1

(U= UL)

4

(U= UR)

2

(U= UL)

3

(U= UR)

Wave pattern of the solution for the Riemann problem


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NCP Flux

Main steps in derivation of NCP flux:

Consider the nonconservative hyperbolic system:

tU + xF(U) + G(U)xU = 0,

Introduce the averaged exact solution ULR(T) as:

ULR(T) =

1

T(SR SL)T SR

T SL

U(x, T) dx.

Apply the Gauss theorem over each subdomain 1, ,4 and connect eachsubdomain using the generalized Rankine-Hugoniot relations.


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The NCP-flux is then given by:

Pnci (UL, UR, v, n

L) =

FLiknLk 12

10 Gikr((; UL, UR))

r

(; UL, UR) dnLk

if SL > v,

{{Fik}}nLk +

12

(SR v)Ui + (SL v)Ui SLULi SRURi )

if SL < v < SR,

FRiknLk + 12 10 Gikr((; UL, UR))r (; UL, UR) dnLkif SR < v,

Note, if G is the Jacobian of some flux function Q, then Pnc(UL, UR, v, nL) isexactly the HLL flux derived for moving grids in van der Vegt and van der Ven

(2002).


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Efficient Solution of Nonlinear Algebraic System

The space-time DG discretization results in a large system of nonlinear algebraicequations:

L(Un; Un1) = 0

This system is solved by marching to steady state using pseudo-time integration

and multigrid techniques:

U

= 1

tL(U; Un1)


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Depth averaged two-fluid model

The dimensionless depth-averaged two fluid model of Pitman and Le, ignoringsource terms for simplicity, can be written as:

tU + xF + GxU = 0,

where:

U =

h(1 )

hhv

hu(1 )b

, F =

h(1 )uhv

hv2 + 12(1 )xxgh2

hu2 + 12gh2

0

G(U) =

0 0 0 0 00 0 0 0 0

gh gh 0 0 (1)xxgh + gh

2u21 u

2 gh gh u2 u( 1) u 2u1 (1 )gh0 0 0 0 0

.


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0 5 10 15 200

0.5

1

1.5

2

x

h

(1)+b,h

+b,h

+b,b

h+b

h(1)+b

h+b

b

Steady-state solution for a subcritical two-phase flow (320 cells).

Total flow height h + b, flow height due to the fluid phase h(1 ), flow height dueto solids phase h and the topography b.


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STDGFEM

h(1) + b h + b

Ncells L2 error p Lmax error p L2 error p Lmax error p

40 0.8171 103 - 0.2308 102 - 0.1404 102 - 0.4194 102 -80 0.2025 103 2.0 0.5584 103 2.0 0.3537 103 2.0 0.9903 103 2.1

160 0.4871 104 2.1 0.1322 103 2.1 0.8511 104 2.1 0.2306 103 2.1320 0.9789 105 2.3 0.2651 104 2.3 0.1712 104 2.3 0.4597 104 2.3

hu(1 ) hv()

Ncells L2 error p Lmax error p L2 error p Lmax error p

40 0.3672 104 - 0.1442 103 - 0.1212 104 - 0.3409 104 -80 0.5911 105 2.6 0.3448 104 2.1 0.1791 105 2.8 0.8054 105 2.1

160 0.1049 105 2.5 0.8471 105 2.0 0.3807 106 2.2 0.2048 105 2.0320 0.1723 106 2.6 0.2078 105 2.0 0.5115 107 2.9 0.4861 106 2.1

Error in h(1) + b, h+ b, hu(1) and hv for subcritical flow over a bump.


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Two-phase dam break problem

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

x

h(1),h,h

h(1) 10000

h 10000

h 10000

h(1) 128

h 128

h 128

(a) Solution of h(1 ), h, band h.

0 0.5 10.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

x

hu(1),hv

hu(1) 10000

hv 10000

hu(1) 128

hv 128

(b) Solution ofhu(1) and hv.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

10000

128

(c) Solution of.

Two-phase dam break problem at time t = 0.175; mesh with 128 elements comparedto mesh with 10000 elements.


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Effect of Path

0 0.2 0.4 0.6 0.8 10.5

0

0.5

1

1.5

2

2.5

3

3.5

x

h(1),h,b,h

h(1) 1

h 1

h 1

h(1) t1

h t1

h t1

h(1) t2

h t2

h t2

h(1) t3

h t3

h t3

h(1) t4

h t4

h t4

h(1) t5

h t5

h t5

b

(d) The solution on the whole domain.

0.095 0.1 0.105 0.11

1.5

2

2.5

3

x

h

(1),h,h

h(1) 1

h 1

h 1

h(1) t1

h t1

h t1

h(1) t2

h t2

h t2

h(1) t3

h t3

h t3h(1) t4

h t4

h t4

h(1) t5

h t5

h t5

(e) The solution zoomed in on the left shock wave.

Solution of h(1

), h, b and h at time t = 0.175 calculated on a mesh with

1024 elements using the paths defined by Toumi.


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Flow Through a Contraction

x1

0

5

10

15

x2

-0.5

0

0.5

h1

Y

X

Z

h: 0.4 0 .5 0.6 0 .7 0.8 0 .9 1 1.1 1 .2 1.3 1 .4 1.5

Flow depth h of water-sand mixture in a contraction

h/L = 0.01, f/s = 0.5, slope 10.


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Compressible Navier-Stokes Equations

Compressible Navier-Stokes equations in space-time domain E:

Ui

x0+Fek (U)

xk F

ek (U,U)xk

= 0

Conservative variables U R5 and inviscid fluxes Fe R53

U =

ujE

, Fek = ukujuk +pjk

huk


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Viscous flux Fv R53

Fv

k =

0jkkj uj qk

with the total stress tensor is defined as:

jk = u

ixi

jk + (u

jxk

+u

kxj

)

and heat flux vector q is defined as:

qk = T

xk


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The viscous flux Fv is homogeneous with respect to the gradient of theconservative variables U:

Fvik(U,U) = Aikrs(U)Ur

xs

with the homogeneity tensor A R

5

3

5

3

defined as:

Aikrs(U) :=Fvik(U,U)

(U)

The system is closed using the equations of state for an ideal gas.


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First Order System

Rewrite the compressible Navier-Stokes equations as a first-order system using theauxiliary variable :

Ui

x0+Feik(U)

xk ik(U)

xk= 0,

ik(U) Aikrs(U)Urxs

= 0.


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Weak Formulation

Weak formulation for the compressible Navier-Stokes equations

Find a U Wh, Vh, such that for all W Wh and V Vh, the followingholds:

KTh

K

Wi

x0Ui +

Wi

xk(Feik ik)

dK

+

KTh

K

WL

i (Ui + Feik ik)nLk d(K) = 0,

KTn

h

K

Vikik dK =

KTnh

K

VikAikrsUr

xsdK

+ KTn

h

Q

VLik ALikrs(Ur ULr )nLs dQ


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Transformation to Arbitrary Lagrangian Eulerian form

The space-time normal vector on a grid moving with velocity v is:

n =

(1, 0, 0, 0)T at K(tn+1),

(1, 0, 0, 0)T at K(t+n ),(vknk, n)T at Qn.

The boundary integral then transforms into:KTh

K

WL

i (Ui + Feik ik)nLk d(K)

=

KTh

K(tn+1)

WL

iUi dK +

K(t+n )

WL

iUi dK

+ KTh

Q

WLi (Feik Uivk ik)nLk dQ


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Numerical Fluxes

The numerical flux U at K(tn+1) and K(t+n ) is defined as an upwind flux toensure causality in time:

U = UL at K(tn+1),

UR at K(t+n

),

At the space-time faces Q we introduce the HLLC approximate Riemann solver asnumerical flux:

nk(

Feik

Uivk)(UL, UR) = H

HLLCi (U

L, UR, v, n)


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ALE Weak Formulation

The ALE flux formulation of the compressible Navier-Stokes equations transformsnow into:

Find a U Wh, such that for all W Wh, the following holds:

KTn

h

KWix0

Ui + Wixk

(Feik ik) dK+

KTnh

K(tn+1)

WL

i ULi dK

K(t+n )

WL

i URi dK

+ KTnhQ WLi (HHLLCi (UL, UR, v, n) iknLk ) dQ = 0.


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Lifting Operator

Introduce the global lifting operator R R53, defined in a weak sense as:Find an R Vh, such that for all V Vh:

KTnh

K

VikRik dK =

SSnI

S

{{VikAikrs}}[[Ur]]s dS

+ SSn

B

S

VLik ALikrs(ULr Ubr )nLs dS.

The weak formulation for the auxiliary variable is now transformed into

KTnh KVikik dK = KTnh K

Vik(AikrsUr

xsRik) dK, V Vh.


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Equation

The primal formulation can be obtained by eliminating the auxiliary variable using

ik = AikrsUr

xsRik, a.e. in Enh

Note, this is possible since hWh Vh


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ALE Weak Formulation for Primal Variables

Recall the ALE flux formulation of the compressible Navier-Stokes equations:

Find a U Wh, such that for all W Wh, the following holds:

KTnh K Wi

x0Ui +

Wi

xk(F

eik

ik) dK

+

KTnh

K(tn+1)

WLi ULi dK

K(t+n )

WLi URi dK

+

KTnh

Q

WLi (HHLLCi (U

L, UR, v, n)

ikn

Lk ) dQ = 0.


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Numerical Fluxes for

The numerical flux in the primary equation is defined following Brezzi as acentral flux = {{}}:

ik(U

L, UR) =

{{Aikrs

Urxs

RSik}} for internal faces,Abikrs

Ubrxs

RSik for boundary faces,

The local lifting operator RS R53 is defined as follows:Find an RS Vh, such that for all V Vh:

KTnh K VikRSik dK =

S

{{VikAikrs}}[[Ur]]s dS for internal faces,

S

VLik ALikrs(U

Lr Ubr )ns dS for external faces.


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Delta Wing Simulations

Simulations of viscous flow about a delta wing with 85 sweep angle.

Conditions

Mach number M = 0.3

Reynolds number Re = 40.000 Angle of attack = 12.5.

Fine grid mesh 1.600.000 elements, 40.000.000 degrees of freedom

Adapted mesh, initial mesh 208.896 elements, after four adaptations 286.416

elements


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Streaklines and vorticity contours in various cross-sections


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Impression of the vorticity based mesh adaptation


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Large eddy simulation of turbulent flow about a delta wing


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Computational Efficiency

Computational efficiency is the key factor limiting industrial applications of higherorder accurate discontinuous Galerkin methods in computational fluid dynamics.

Multigrid methods are good candidates to increase computational efficiency, butneed significant improvements for higher order accurate DG discretizations.


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Objectives

Perform a theoretical analysis of multigrid performance for advection dominatedflows, in particular for higher order accurate DG discretizations.

Improve multigrid performance using theoretical analysis tools.

Test multigrid performance on realistic problems.


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Main Components of h-Multigrid Algorithm

Consider a finite sequence Nc of increasingly coarser meshes Mnh,n {1, , Nc}

Define operators to connect data on the different meshes:

restriction operators

Rmhnh : Mnh Mmh, 1 n < m Nc,

prolongation operators

Pnh

mh : Mmh Mnh, 1 n < m Nc.


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Use iterative solvers Snh to approximately solve the system of algebraic equationson the various grid levels

Lnhvh = fnh on Mnh

Since, the main eff

ect of the multigrid algorithm should be the damping of highfrequency error components Snh is called a smoothing operator.

Choose a cycling strategy between the different meshes, e.g. V- or W-cycle.


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Multigrid error transformation operator for linear problems

In order to understand the performance of the multigrid algorithm we need toinvestigate the multigrid error transformation operator.

The multigrid error transformation operator shows how much the error in theiterative solution of the algebraic system is reduced by one full multigrid cycle.

We analyze three-level multigrid algorithms for 2D problems to obtain betterestimates for the convergence rate.


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Given an initial error eAh , the error eDh after one full multigrid cycle with three gridlevels is given by the relation

eDh = M

3gh e

Ah

with

M3gh = S2h (Ih Ph2h(I2h Mc2h)L12h R2hh Lh)S1hand

M2h = S42h(I2h P2h4h L14h R4h2hL2h)S

32h.

The properties of the multigrid error transformation operator are analyzed usingdiscrete Fourier analysis.


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Three-grid Fourier analysis

/2 /2 0

0

/2

/2

2

1

/4

/4

/4

/4

Low, medium and high frequencies Fourier modes in three-level multigrid


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Multigrid Error Transformation Operator

The discrete Fourier transform of the error transformation operator for a three-levelmultigrid cycle M3gh () C16m16m which is equal to

M3gh () =

S3gh ()

2

I3g

P3gh ()

U3g(; c)

Q3gh ()

R3gh ()

L3gh ()

S3gh ()

1

4h \3g

The matrix U3g(; c) is equal toU3g(; c) = I2g M2g2h(2)c.

with

M2g2h(2) = S2g2h(2)4I2gP2g2h (2)L14h (400)R2g2h(2)L2g2h(2)S2g2h(2)3


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Asymptotic Convergence Rate

The asymptotic convergence factor per cycle is defined as

= limm

sup

e(0)

h =0

e(m)h 2(Gh)

e

(0)h

2(G

h)

1m

and can be rewritten into

= (Mngh ).


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Optimization of multigrid performance

The spectral radius of the error transformation operator can be used to optimizethe multigrid algorithm.

In particular, the smoother is a good candidate for optimization since in general itcontains a number of free parameters.

We will consider a Runge-Kutta type smoother.


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Analysis and Optimization of Runge-Kutta Smoothers for theAdvection-Diffusion Equation

The advection-diffusion equation in the domain is defined as

u

t+ a u = (Au))

The equations are discretized with a higher order accurate space-timediscontinuous Galerkin finite element method.

The space-time discretizations are solved using a multigrid algorithm with aRunge-Kutta type smoother.


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Pseudo-time Runge-Kutta smoothers

Let the system of algebraic equations be denoted as

L(un, un1) = 0.

A pseudo time derivative is added to the system and integrated to steady-state in

pseudo-time

Mu

= L(u, un1),

At steady state un = u.

For the pseudo-time integration we consider 4- and 5-stage Runge-Kutta methods.


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A four-stage Runge-Kutta scheme is given by:

(1 + 1I)V1

=V0 1

xy21M

1L(V0) + 1V0

(1 + 2I)V2

=V0 M

1

xy

31L(V

0) + 32L(V

1)

+ 2V1

(1 + 3I)V3 =V0 M

1

xy41L(V0) + 42L(V1) + 43L(V2) + 3V2

(1 + 4I)V4

=V0 M

1

xy

51L(V

0) + 52L(V

1) + 53L(V

2) + 54L(V

3)

+4V3

We require from the 4- and 5-stage Runge-Kutta schemes that they are secondorder accurate


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Optimized Runge-Kutta smoothers for space-time DGFEM

Optimization of the smoother coefficients for three-level multigrid algorithms

Space-time DG discretization of the 2D advection-diffusion equation usingquadratic basis-functions

For the optimization process, we fix the Reynolds numbers Rex = Rey = 100,the flow angle flow = /4 and the aspect ratio AR = 1.

For steady flows we fix CF Lt = 100, while for unsteady flows CF Lt = 1.


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The optimization process has a big impact on the Runge-Kutta coefficients whichgreatly differ per case.

The Runge-Kutta coefficients are very different from commonly used schemes sincethey are optimized for fast multigrid convergence and not for time accuracy.

Finding a good balance between optimization and more generally applicablealgorithms is still an open question.

In practice, the Runge-Kutta smoothers use local time stepping and this gives theopportunity to apply locally the best smoother for the actual flow state.


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Optimized coeffi

cients for dRK5 and fRK5 smoothers for 3-level multigrid (steady flow).dRK5 p = 1 fRK5 p = 1 dRK5 p = 2 fRK5 p = 2

21 0.05768995298 0.0578331573 0.04865009589 0.0487743632531 - -0.0002051554736 - -0.000218834843832 0.1405960888 0.1403808301 0.130316854 0.130090612241 - 0.0003953470071 - 2.608884832e-0542 - -0.001195029164 - 2.444376496e-0543 0.267958213 0.2681810517 0.2729621396 0.273480570551 - 0.0001441249202 - -0.00125038548752 - -0.0002608610327 - -0.0007838720635

53 - -0.0003368070181 - -0.000489088771254 0.5 0.8473374098 0.5 4.41213936761 - 0.4115573097 - 0.809721735862 - -0.003144851878 - 0.0843508900963 - -0.0001096455683 - -0.0198679900764 - 0.001555741114 - 0.0135981547665 1.0 0.5901414466 1.0 0.1121972094CFL 0.8 0.8 0.4 0.4

S 0.98812 0.98914 0.98974 0.9896

M G 0.89151 0.81762 0.90049 0.89903

EX IMG 167.06 - 124.02 -


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10 8 6 4 2 0 25

4

3

2

1

0

1

2

3

4

5

Stability domain of smoother

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.20.2

0.2

0.3

0.3

0.3

0.3

0.3

0.3

0.4

0.40

.4

0.4

0.4

0.4

0.5

0.50

.5

0.5

0.5

0.5

0.6

0.6

0.6

0.6

0.6

0.6

0.7

0.7

0.7

0.7

0.7

0.7

0.7

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.9

0.90.9

0.9

0.9

0.9

0.9

1

11

1

1

1

1

Re(z)

Im(z)

(a) Stability domain dRK5.

10 8 6 4 2 0 25

4

3

2

1

0

1

2

3

4

5

Re(z)

Im(z)

Spectrum of L and stability domain of smoother

low

high

(b) Spectrum of Lh and stability domain dRK5.

Stability domain of dRK5 smoother and spectrum of a space-time DG discretization of

the 2D advection-diffusion equation using quadratic basis functions.


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10 8 6 4 2 0 25

4

3

2

1

0

1

2

3

4

5

Stability domain of smoother

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.2

0.20.2

0.3

0.3

0.3

0.3

0.3

0.3

0.4

0.40

.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.5

0.5

0.6

0.6

0.6

0.6

0.6

0.6

0.7

0.7

0.7

0.7

0.70.7

0.8

0.8

0.8

0.

8

0.80.8

0.8

0.9

0.90.9

0.9

0.90.9

0.9

1

11

1

11

1

Re(z)

Im(z)

(a) Stability domain fRK5.

10 8 6 4 2 0 25

4

3

2

1

0

1

2

3

4

5

Re(z)

Im(z)

Spectrum of L and stability domain of smoother

low

high

(b) Spectrum ofLh and stability domain fRK5.

Stability domain of the fRK5 smoother and spectrum of a space-time DG discretization

of the 2D advection-diffusion equation using quadratic basis functions.


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1 0.5 0 0.5 1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

Re

Im

Spectrum of smoother

low

high

(a) dRK5 smoother.

1 0.5 0 0.5 1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

Re

Im

Spectrum of smoother

low

high

(b) fRK5 smoother.

1 0.5 0 0.5 1

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Re

Im

Spectrum of threelevel operator

(c) 3-level MG with dRK5.

1 0.5 0 0.5 1

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Re

Im

Spectrum of threelevel operator

(d) 3-level MG with fRK5.

Eigenvalue spectra for space-time DG discretizations of the 2D advection-diffusion

equation using quadratic basis functions.


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Testing multigrid performance

In order to demonstrate the performance of the optimized algorithms we considerthe 2D advection-diffusion equation on a square.

The exact steady state solution is

u(x1, x2) =1

2

exp(a1/1) exp(a1x1/1)exp(a1/1) 1

+exp(a2/2) exp(a2x2/2)

exp(a2/2) 1

.


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The advection-diffusion equation is discretized using the space-time discontinuousGalerkin discretization with quadratic basis functions.

In the discretization we use a Shishkin mesh which is suitable for dealing withboundary layers.

In pseudo-time we apply a rescaling to reduce the effect of grid stretching.


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The parameters in the tests are:

t = 100, a =

2, x = y = 0.01, N1 = N2 = 32.

Flow angle flow = /4.

Depending on the stability of the smoother, we use different CF L and V N

numbers.

The EXV scheme was used when Rei 1 and the EXI scheme was usedotherwise.

For the multigrid computations, 1 = 2 = 3 = 4 = 1, C = 4 and = 1.


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Work units

L

/L0

500 1000 1500 2000 2500 300010

-4

10-3

10-2

10-1

dRK5 coarse a pprox.

fRK5 coarse approx.EXI/EXV coarse approx.

dRK5 coarse exact

fRK5 coarse exactEXI/EXV coarse exact

Convergence of three-level multigrid with different Runge-Kutta smoothers for secondorder accurate space-time DG discretization.


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We can draw the following conclusions:

In all cases using the optimized Runge-Kutta smoothers a big improvement isobtained over the original EXI-EXV smoother.

For linear basis functions the number of multigrid work units to obtain 4 orders of

reduction in the residual is reduced from 3300 to 380.

For quadratic basis functions the number of multigrid work units reduces from22000 to 184.

For linear basis functions the difference in convergence rate between Runge-Kuttasmoothers with only non-zero diagonal terms versus Runge-Kutta smoothers with

all coefficients non-zero is negligible.


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For quadratic basis functions this difference is, however, significant.

Using more Runge-Kutta coefficients enlarges the possibilities to optimize thesmoother, but the optimization process requires a significantly larger computing

time.

In order to speed up the optimization process the coefficients of Runge-Kuttaschemes with only non-zero diagonal terms are used as initial values.


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The effect of solving the equations on the coarsest mesh is very large.

In particular, for nonlinear problems it is tempting to solve the algebraic system onthe coarsest mesh only approximately, but the effect is non-negligible.

The flow angle has a small effect on the convergence rate. In general if the flow

angle is close to one of the mesh lines the convergence rate is the slowest.

It is important not to extrapolate the results for the advection-diffusion equationdirectly to the Euler and Navier-Stokes equations.


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Euler Equations

Amount of work per multigrid cycle differs single grid, p-, h-, and hp-multigrid:

work per cycle =

gpbp, SG,

(gpbp + gp1bp1)(p1 + p2 ) + g

p2bp2pC, pMG,

gpbp

(ch + ch1)(h1 + h2 ) + c

h2hC

, hMG,

ch(gpbp + gp1bp1)(p1 + p2 )+

gp2bp2((ch + ch1)(h1 + h2 ) + ch2hC), hpMG,

with gp and bp, respectively, the number of Gauss quadrature points and basis

functions in an element, and ch a weighting for the number of cells depending on the

grid-level h.

Coefficients: gp = 9, gp1 = 4, gp2 = 1, bp = 6, bp1 = 4, bp2 = 1, ch = 1,ch1 = 1/4 and ch2 = 1/16.

U i i f T Ch i N i l A l i d C i l M h i


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p-levels h-levels CFL work

1 1 1.0 0.99010 75006

3 1 1.0 0.74318 33370

3 3 1.0 0.73347 32091

1 3 1.0 0.66350 24098

1 3 0.5 0.80364 45360

Spectral radii and multigrid work units of different multigrid strategies.

U i i f T Ch i N i l A l i d C i l M h i 78


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1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0 200 400 600 800 1000 1200 1400 1600

L2

total

Work Units

SGhMG

hMG exactpMG

hpMG

Convergence history of different multigrid techniques for space-time DG discretization

for inviscid flow around a NACA0012 airfoil (MTC 1 test case, = 2, M a = 0.5,448 64 elements).

U i it f T t Ch i N i l A l i d C t ti l M h i 79


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Conclusions

A space-time DG discretization for nonconservative hyperbolic pdes using theDLM theory and the compressible Navier-Stokes has been developed.

A new numerical flux for nonconservative hyperbolic pdes has been developed,which reduces to the HLLC flux for conservative pdes.

The effect of the choice of the path in phase space is in practice for nearly all casesnegligible.

The algorithm has been successfully tested on a depth averaged two-phase flowmodel and compressible flow simulations.

U i it f T t Ch i N i l A l i d C t ti l M h i 80


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Conclusions

A detailed three-level multigrid analysis has been conducted for h-, p- andhp-multigrid algorithms for the advection-diffusion and the linearized Euler

equations.

The analysis provides the asymptotic convergence rate of the multigrid algorithms

which is used to optimize the multigrid smoothers.

For the advection-diffusion equation the optimization results in a significantimprovement in the convergence rate in actual computations using the h-multigrid

algorithm.

University of Twente Chair Numerical Analysis and Computational Mechanics 81


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For the Euler equations the optimized Runge-Kutta smoothers for h-, p- andhp-multigrid show initially a significant improvement in convergence rate for

inviscid flow about a NACA 0012 airfoil, but asymptotically most algorithms have

the same convergence rate.

Using an exact solution of the algebraic system on the coarsest mesh has a major

impact ion the convergence rate for the advection-diffusion equation, but not forthe Euler equations.

Vander Vegt

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