Discontinuous Galerkin Finite Element Methods for ... · Discontinuous Galerkin Finite Element Methods for Compressible Flows Jaap van der Vegt Numerical Analysis and ... • Algorithms

Discontinuous Galerkin Finite Element Methods for

Compressible Flows

Jaap van der Vegt

Numerical Analysis and Computational Mechanics Group

Department of Applied Mathematics

University of Twente

Joint Work with

Sander Rhebergen, Onno Bokhove, Christiaan Klaij, Fedderik van der Bos

Toulouse, December 8, 2009

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 are

suitable for both conservative and nonconservative partial differential equations


Introduction

Motivation of research:

• Nonconservative hyperbolic partial differential equations contain nonconservative

products

∂xu + A(u)∂xu = 0

• The essential feature of nonconservative products is that A 6= Df , hence A is

not the Jacobian matrix of a flux function f .

• This causes problems once the solution becomes discontinuous, because the weak

solution in the classical sense of distributions then does not exist.

• This also complicates the derivation of discontinuous Galerkin discretizations since

there is no direct link with a Riemann problem.


• Alternative: use the theory for nonconservative products from Dal Maso, LeFloch

and Murat (DLM)


Overview of Presentation

• Overview of main results of the theory of Dal Maso, LeFloch and Murat for

nonconservative products

• Space-time DG discretization of nonconservative hyperbolic partial differential

equations

• Extension to the compressible Navier-Stokes equations

• Multigrid techniques for space-time DG discretizations

• Conclusions


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 → R

m the product g(u)∂xu is not defined at

x = xd since here |∂xu| → ∞.

• Introduce a smooth regularization uε of u. If the total variation of uε remains

uniformly bounded with respect to ε then Dal Maso, LeFloch and Murat (DLM)

showed that

g(u)du

dx≡ lim

ε→0g(u

ε)duε

dxgives a sense to the nonconservative product as a bounded measure.


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 φ : [0, 1] → Rm, satisfying φ(0) = uL

and φ(1) = uR, connecting uL and uR in Rm.

• The following regularization uε for u then emerges:

uε(x) =

uL, if x ∈]a, xd − ε[,

φ(x−xd+ε

2ε ), if x ∈]xd − ε, xd + ε[, ε > 0.

uR, if x ∈]xd + ε, b[


• When ε tends to zero, then:

g(uε)duε

dx Cδxd

, with C =

∫ 1

0

g(φ(τ))dφ

dτ(τ) dτ,

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

• The limit of g(uε)∂xuε depends on the path φ.

• There is one exception, namely if an q : Rm → R exists with g = ∂uq. In this

case C = q(uR) − q(uL).


DLM Theory

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

hyperbolic pde’s.

• Introduce the Lipschitz continuous maps φ : [0, 1] × Rm × R

m → Rm which

satisfy the following properties:

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

(H2) φ(τ ; uL, uL) = uL,

(H3)∣∣ ∂φ

∂τ (τ ; uL, uR)∣∣ ≤ K|uL − uR|, a.e. in [0, 1].


• Theorem (DLM). Let u :]a, b[→ Rm be a function of bounded variation and

g : Rm → R

m a continuous function. Then, there exists a unique real-valued

bounded Borel measure µ on ]a, b[ with:

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

µ(B) =

∫

B

g(u)du

dx

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

µ(xd) =

∫ 1

0

g(φ(τ ; uL, uR))∂φ

∂τ(τ ; uL, uR) dτ.

By definition, this measure µ is the nonconservative product of g(u) by ∂xu and

denoted by µ =[g(u)du

dx

]φ

.


Rankine-Hugoniot Relations

• For conservative hyperbolic system of pde’s, ∂xu + ∂xf(u) = 0 the

Rankine-Hugoniot relations across a jump with uL and uR and velocity v are

equal to

−v(uR − u

L) + f(u

R) − f(u

L) = 0.

• For a nonconservative hyperbolic pde ∂xu + A(u)∂xu = 0 the Rankine-Hugoniot

relations in the DLM theory are equal to

−v(uR − u

L) +

∫ 1

0

A(φD(s, uL, u

R))∂sφD(s; u

L, u

R)ds = 0

with φD a Lipschitz continuous path satisfying φD(0; uL, uR) = uL and

φD(1; uL, uR) = uR.

• The Rankine-Hugoniot relations are essential for the definition of the NCP flux

used in the DG discretization.


Space-Time Discontinuous Galerkin Method

I nt n+1

tn

t0

Qn

Ω

Ω n

n+1

y

x

tK

• A time-dependent problem is considered directly in four dimensional space, with

time as the fourth dimension


Key Features of Space-Time Discontinuous Galerkin Methods

• Simultaneous discretization in space and time: time is considered as a fourth

dimension.

• Discontinuous basis functions, both in space and time, with only a weak coupling

across 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.


Benefits of Discontinuous Galerkin Methods

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

I achieving higher order accuracy on unstructured meshes

I hp-mesh adaptation

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

hexahedra and prisms

I parallel computing


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 free

boundaries and after mesh adaptation.


Disadvantages of Space-(Time) Discontinuous Galerkin Methods

• Algorithms are generally rather complicated, in particular for elliptic and parabolic

partial differential equations

• On structured meshes DG methods are computationally more expensive than finite

difference and finite volume methods.


Space-Time Domain

• Consider an open domain: E ⊂ Rd.

• The flow domain Ω(t) at time t is defined as:

Ω(t) := x ∈ E | x0 = t, t0 < t < T

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

Ω(t0) :=x ∈ ∂E | x0 = t0,

Ω(T ) :=x ∈ ∂E | x0 = T,

Q :=x ∈ ∂E | t0 < x0 < T.

• The space-time domain is covered with a tessellation Th consisting of space-time

elements K.


Discontinuous Finite Element Approximation

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

Wh :=

W ∈ (L2(Eh))m : W |K GK ∈ (P k(K))m, ∀K ∈ Th

• The jump of f at an internal face S ∈ SnI in the direction k of a Cartesian

coordinate system is defined as:

[[f ]]k = fLnLk + fRnR

k ,

with nRk = −nL

k .

• The average of f at S ∈ SnI is defined as:

f = 12(f

L+ f

R).


Space-Time DG Formulation of Nonconservative HyperbolicPDE’s

• Consider the nonlinear hyperbolic system of partial differential equations in

nonconservative form in multi-dimensions:

∂Ui

∂t+

∂Fik

∂xk

+ Gikr

∂Ur

∂xk

= 0, x ∈ Ω ⊂ Rq, t > 0,

with U ∈ Rm, F ∈ R

m × Rq, G ∈ R

m × Rq × R

m

• These equations model for instance bubbly flows, granular flows, shallow water

equations and many other physical systems.


• Weak formulation for nonconservative hyperbolic system:

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

∑

K∈Th

∫

KVi

(Ui,0 + Fik,k + GikrUr,k

)dK

+∑

K∈Th

( ∫

K(t−n+1)

Vi(URi − UL

i ) dK −∫

K(t+n )

Vi(URi − UL

i ) dK

)

+∑

S∈SI

∫

SVi

( ∫ 1

0

Gikr(φ(τ ; UL, U

R))

∂φr

∂τ(τ ; U

L, U

R) dτ n

Lk

)dS

−∑

S∈SI

∫

SVi[[Fik − vkUi]]k dS = 0


Relation with Space-Time DG Formulation of ConservativeHyperbolic PDE’s

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

V =

V at S ∈ SI,

0 at K(tn) ⊂ Ωh(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.


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

∑

K∈Th

∫

K

(− Vi,0Ui − Vi,kFik + ViGikrUr,k

)dK

+∑

K∈Th

( ∫

K(t−n+1)

V Li UL

i dK −∫

K(t+n )

V Li UL

i dK

)

+∑

S∈SI

∫

S(V

Li − V

Ri )Fik − vkUin

Lk dS

+∑

S∈SB

∫

SV L

i (F Lik − vkU

Li )nL

k dS

+∑

S∈SI

∫

SVi

( ∫ 1

0

Gikr(φ(τ ; UL, U

R))

∂φr

∂τ(τ ; U

L, U

R) dτ n

Lk

)dS = 0


Numerical Fluxes

• The fluxes at the element faces do not contain any stabilizing terms yet, both for

the conservative and nonconservative part

• At the time faces, the numerical flux is selected such that causality in time is

ensured

U =

UL at K(t−n+1)

UR at K(t+n )

.

• The space-time DG formulation is stabilized using the NCP (Non-Conservative

Product) flux

Pnci =

(Fik − vkUi + Pik

)n

Lk


Nonconservative Product Flux

-

6

@@

@@

@@

@@

@@

@@

@@@

xL xR

x

t

SL SRSM

T

Ω1

(U = UL)

Ω4

(U = UR)

Ω2

(U = U∗

L)

Ω3

(U = U∗

R)

Wave pattern of the solution for the Riemann problem


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 U∗LR(T ) as:

U∗LR(T ) =

1

T (SR − SL)

∫ TSR

TSL

U(x, T ) dx.

• Apply the Gauss theorem over each subdomain Ω1, · · · , Ω4 and connect each

subdomain using the generalized Rankine-Hugoniot relations.


• The NCP-flux is then given by:

P nci (UL, UR, v, nL) =

F LiknL

k − 12

∫ 10 Gikr(φ(τ ; UL, UR))∂φr

∂τ (τ ; UL, UR) dτnLk

if SL > v,

FiknLk + 1

2

((SR − v)U∗

i + (SL − v)U∗i − SLUL

i − SRURi )

if SL < v < SR,

F RiknL

k + 12

∫ 10 Gikr(φ(τ ; UL, UR))∂φr

∂τ (τ ; UL, UR) dτnLk

if SR < v,

• Note, if G is the Jacobian of some flux function Q, then P nc(UL, UR, v, nL) is

exactly the HLL flux derived for moving grids in van der Vegt and van der Ven

(2002).


Efficient Solution of Nonlinear Algebraic System

• The space-time DG discretization results in a large system of nonlinear algebraic

equations:

L(Un; Un−1) = 0

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

and multigrid techniques:

∂U

∂τ= − 1

∆tL(U ; U

n−1)


Depth averaged two-fluid model

• The dimensionless depth-averaged two fluid model of Pitman and Le, ignoring

source terms for simplicity, can be written as:

∂tU + ∂xF + G∂xU = 0,

where:

U =

h(1 − α)hαhαv

hu(1 − α)b

, F =

h(1 − α)uhαv

hαv2 + 12ε(1 − ρ)αxxgh2α

hu2 + 12εgh2

0

G(U) =

0 0 0 0 00 0 0 0 0

εραgh εραgh 0 0 ε(1 − ρ)αxxghα + εραgh

2u2α1−α − αu2 − εghα −εghα − αu2 u(α − 1) uα − 2uα

1−α (1 − α)εgh

0 0 0 0 0

.


0 5 10 15 200

0.5

1

1.5

2

x

h(1−

α)+

b,hα

+b,

h+b,

b

h+bh(1−α)+bhα+bb

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 − α), flow height due

to solids phase hα and the topography b.


STDGFEM

h(1 − α) + b hα + b

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

40 0.8171 · 10−3 - 0.2308 · 10−2 - 0.1404 · 10−2 - 0.4194 · 10−2 -

80 0.2025 · 10−3 2.0 0.5584 · 10−3 2.0 0.3537 · 10−3 2.0 0.9903 · 10−3 2.1

160 0.4871 · 10−4 2.1 0.1322 · 10−3 2.1 0.8511 · 10−4 2.1 0.2306 · 10−3 2.1

320 0.9789 · 10−5 2.3 0.2651 · 10−4 2.3 0.1712 · 10−4 2.3 0.4597 · 10−4 2.3

hu(1 − α) hv(α)

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

40 0.3672 · 10−4 - 0.1442 · 10−3 - 0.1212 · 10−4 - 0.3409 · 10−4 -

80 0.5911 · 10−5 2.6 0.3448 · 10−4 2.1 0.1791 · 10−5 2.8 0.8054 · 10−5 2.1

160 0.1049 · 10−5 2.5 0.8471 · 10−5 2.0 0.3807 · 10−6 2.2 0.2048 · 10−5 2.0

320 0.1723 · 10−6 2.6 0.2078 · 10−5 2.0 0.5115 · 10−7 2.9 0.4861 · 10−6 2.1

Error in h(1−α)+ b, hα+ b, hu(1−α) and hvα for subcritical flow over a bump.


Two-phase dam break problem

0 0.5 10

0.5

1

1.5

2

2.5

3

3.5

x

h(1−

α),h

α,h

h(1−α) 10000hα 10000h 10000h(1−α) 128hα 128h 128

(a) Solution of h(1 − α), hα, band h.

0 0.5 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

x

hu(1

−α)

,hvα

hu(1−α) 10000hvα 10000hu(1−α) 128hvα 128

(b) Solution of hu(1−α) and hvα.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

x

α

α 10000α 128

(c) Solution of α.

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

to mesh with 10000 elements.


Effect of Path

0 0.2 0.4 0.6 0.8 1−0.5

0

0.5

1

1.5

2

2.5

3

3.5

x

h(1−

α),h

α,b,

h

h(1−α) 1hα 1h 1h(1−α) t1hα t1h t1h(1−α) t2hα t2h t2h(1−α) t3hα t3h t3h(1−α) t4hα t4h t4h(1−α) t5hα t5h t5b

(d) The solution on the whole domain.

0.095 0.1 0.105 0.11

1.5

2

2.5

3

x

h(1−

α),h

α,h

h(1−α) 1hα 1h 1h(1−α) t1hα t1h t1h(1−α) t2hα t2h t2h(1−α) t3hα t3h t3h(1−α) t4hα t4h t4h(1−α) t5hα t5h t5

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

Solution of h(1 − α), hα, b and h at time t = 0.175 calculated on a mesh with

1024 elements using the paths defined by Toumi.


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, ρf/ρs = 0.5, slope 10.


Compressible Navier-Stokes Equations

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

∂Ui

∂x0

+∂F e

k(U)

∂xk

− ∂F ek(U,∇U)

∂xk

= 0

• Conservative variables U ∈ R5 and inviscid fluxes F e ∈ R

5×3

U =

ρ

ρuj

ρE

, F ek =

ρuk

ρujuk + pδjk

ρhuk



• Viscous flux F v ∈ R5×3

Fvk =

0

τjk

τkjuj − qk

with the total stress tensor τ is defined as:

τjk = λ∂ui

∂xi

δjk + µ(∂uj

∂xk

+∂uk

∂xj

)

and heat flux vector ~q is defined as:

qk = −κ∂T

∂xk



• The viscous flux F v is homogeneous with respect to the gradient of the

conservative variables ∇U :

Fvik(U,∇U) = Aikrs(U)

∂Ur

∂xs

with the homogeneity tensor A ∈ R5×3×5×3 defined as:

Aikrs(U) :=∂F v

ik(U,∇U)

∂(∇U)

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


First Order System

• Rewrite the compressible Navier-Stokes equations as a first-order system using the

auxiliary variable Θ:

∂Ui

∂x0

+∂F e

ik(U)

∂xk

− ∂Θik(U)

∂xk

= 0,

Θik(U) − Aikrs(U)∂Ur

∂xs

= 0.


Weak Formulation

• Weak formulation for the compressible Navier-Stokes equations

Find a U ∈ Wh, Θ ∈ Vh, such that for all W ∈ Wh and V ∈ Vh, the following

holds:

−∑

K∈Th

∫

K

(∂Wi

∂x0

Ui +∂Wi

∂xk

(F eik − Θik)

)dK

+∑

K∈Th

∫

∂KW

Li (Ui + F

eik − Θik)n

Lk d(∂K) = 0,

∑

K∈T nh

∫

KVikΘik dK =

∑

K∈T nh

∫

KVikAikrs

∂Ur

∂xs

dK

+∑

K∈T nh

∫

QV L

ikALikrs(Ur − UL

r )nLs dQ


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(t−n+1),

(−1, 0, 0, 0)T at K(t+n ),

(−vknk, n)T at Qn.

• The boundary integral then transforms into:

∑

K∈Th

∫

∂KW

Li (Ui + F

eik − Θik)n

Lk d(∂K)

=∑

K∈Th

( ∫

K(t−n+1)

W Li Ui dK +

∫

K(t+n )

W Li Ui dK

)

+∑

K∈Th

∫

QW

Li (F

eik − Uivk − Θik)n

Lk dQ


Numerical Fluxes

• The numerical flux U at K(t−n+1) and K(t+n ) is defined as an upwind flux to

ensure causality in time:

U =

UL at K(t−n+1),

UR at K(t+n ),

• At the space-time faces Q we introduce the HLLC approximate Riemann solver as

numerical flux:

nk(Feik − Uivk)(U

L, UR) = HHLLCi (UL, UR, v, n)


ALE Weak Formulation

• The ALE flux formulation of the compressible Navier-Stokes equations transforms

now into:

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

−∑

K∈T nh

∫

K

(∂Wi

∂x0

Ui +∂Wi

∂xk

(Feik − Θik)

)dK

+∑

K∈T nh

( ∫

K(t−n+1)

WLi U

Li dK −

∫

K(t+n )

WLi U

Ri dK

)

+∑

K∈T nh

∫

QW L

i (HHLLCi (UL, UR, v, n) − Θikn

Lk ) dQ = 0.


Lifting Operator

• Introduce the global lifting operator R ∈ R5×3, defined in a weak sense as:

Find an R ∈ Vh, such that for all V ∈ Vh:

∑

K∈T nh

∫

KVikRik dK =

∑

S∈SnI

∫

SVikAikrs[[Ur]]s dS

+∑

S∈SnB

∫

SV

LikA

Likrs(U

Lr − U

br)n

Ls dS.

• The weak formulation for the auxiliary variable is now transformed into

∑

K∈T nh

∫

KVikΘik dK =

∑

K∈T nh

∫

KVik(Aikrs

∂Ur

∂xs

− Rik) dK, ∀V ∈ Vh.


Θ Equation

• The primal formulation can be obtained by eliminating the auxiliary variable Θ

using

Θik = Aikrs

∂Ur

∂xs

− Rik, a.e. in Enh

• Note, this is possible since ∇hWh ⊂ Vh


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:

−∑

K∈T nh

∫

K

(∂Wi

∂x0

Ui +∂Wi

∂xk

(Feik − Θik)

)dK

+∑

K∈T nh

( ∫

K(t−n+1)

W Li UL

i dK −∫

K(t+n )

W Li UR

i dK)

+∑

K∈T nh

∫

QW L

i (HHLLCi (UL, UR, v, n) − Θikn

Lk ) dQ = 0.


Numerical Fluxes for Θ

• The numerical flux Θ in the primary equation is defined following Brezzi as a

central flux Θ = Θ:

Θik(UL, UR) =

Aikrs

∂Ur∂xs

− ηRSik for internal faces,

Abikrs

∂Ubr

∂xs− ηRS

ik for boundary faces,

• The local lifting operator RS ∈ R5×3 is defined as follows:

Find an RS ∈ Vh, such that for all V ∈ Vh:

∑

K∈T nh

∫

KVikRS

ik dK =

∫

SVikAikrs[[Ur]]s dS for internal faces,

∫

SV L

ikALikrs(U

Lr − Ub

r)ns dS for external faces.


Delta Wing Simulations

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

• Conditions

I Mach number M = 0.3

I Reynolds number Re = 40.000

I Angle of attack α = 12.5.

I Fine grid mesh 1.600.000 elements, 40.000.000 degrees of freedom

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

elements



Streaklines and vorticity contours in various cross-sections



Impression of the vorticity based mesh adaptation



Large eddy simulation of turbulent flow about a delta wing


Computational Efficiency

• Computational efficiency is the key factor limiting industrial applications of higher

order accurate discontinuous Galerkin methods in computational fluid dynamics.

• Multigrid methods are good candidates to increase computational efficiency, but

need significant improvements for higher order accurate DG discretizations.


Objectives

• Perform a theoretical analysis of multigrid performance for advection dominated

flows, in particular for higher order accurate DG discretizations.

• Improve multigrid performance using theoretical analysis tools.

• Test multigrid performance on realistic problems.


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:

I restriction operators

Rmhnh : Mnh → Mmh, 1 ≤ n < m ≤ Nc,

I prolongation operators

Pnhmh : Mmh → Mnh, 1 ≤ n < m ≤ Nc.


• Use iterative solvers Snh to approximately solve the system of algebraic equations

on the various grid levels

Lnhvh = fnh on Mnh

• Since, the main effect of the multigrid algorithm should be the damping of high

frequency error components Snh is called a smoothing operator.

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


Multigrid error transformation operator for linear problems

• In order to understand the performance of the multigrid algorithm we need to

investigate the multigrid error transformation operator.

• The multigrid error transformation operator shows how much the error in the

iterative solution of the algebraic system is reduced by one full multigrid cycle.

• We analyze three-level multigrid algorithms for 2D problems to obtain better

estimates for the convergence rate.


• Given an initial error eAh , the error eD

h after one full multigrid cycle with three grid

levels is given by the relation

eDh = M

3gh e

Ah

with

M3gh = S

ν2h (Ih − P

h2h(I2h − M

γc2h)L

−12h R

2hh Lh)S

ν1h

and

M2h = Sν42h(I2h − P 2h

4h L−14h R4h

2hL2h)Sν32h.

• The properties of the multigrid error transformation operator are analyzed using

discrete Fourier analysis.


Three-grid Fourier analysis

−π −π/2 π/2 π0

0

π/2

π

−π/2

−π

θ2

θ1

π/4

−π/4

π/4

−π/4

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


Multigrid Error Transformation Operator

• The discrete Fourier transform of the error transformation operator for a three-level

multigrid cycle M3gh (θ) ∈ C

16m×16m which is equal to

M3gh (θ) =

(S3g

h (θ))ν2

(I3g−P 3g

h (θ)U3g(θ; γc)Q3gh (θ)R3g

h (θ)L3gh (θ)

)(S3g

h (θ))ν1

∀θ ∈ Θ4h \ Ψ3g

The matrix U3g(θ; γc) is equal to

U3g(θ; γc) = I2g −(M2g

2h(2θβ))γc

.

with

M2g2h(2θβ) =

(S

2g2h(2θβ)

)ν4(

I2g−P

2g2h(2θβ)L

−14h (4θ

00)R

2g2h(2θβ)L

2g2h(2θβ)

)(S

2g2h(2θβ)

)ν3


Asymptotic Convergence Rate

• The asymptotic convergence factor per cycle is defined as

µ = limm→∞

sup

e(0)h

6=0

‖e(m)h ‖`2(Gh)

‖e(0)h ‖`2(Gh)

1m

and can be rewritten into

µ = ρ(Mngh ).


Optimization of multigrid performance

• The spectral radius of the error transformation operator can be used to optimize

the multigrid algorithm.

• In particular, the smoother is a good candidate for optimization since in general it

contains a number of free parameters.

• We will consider a Runge-Kutta type smoother.


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

• The advection-diffusion equation in the domain Ω is defined as

∂u

∂t+ a · ∇u = ∇ · (A∇u))

• The equations are discretized with a higher order accurate space-time

discontinuous Galerkin finite element method.

• The space-time discretizations are solved using a multigrid algorithm with a

Runge-Kutta type smoother.


Pseudo-time Runge-Kutta smoothers

• Let the system of algebraic equations be denoted as

L(un, u

n−1) = 0.

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

pseudo-time

M ∂u∗

∂τ= −L(u∗, un−1),

• At steady state un = u∗.

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


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

(1 + β1λI)V 1 =V 0 − 1

∆x∆yα21λM−1L(V 0) + β1λV 0

(1 + β2λI)V2=V

0 − λM−1

∆x∆y

(α31L(V

0) + α32L(V

1))

+ β2λV1

(1 + β3λI)V 3 =V 0 − λM−1

∆x∆y

(α41L(V 0) + α42L(V 1) + α43L(V 2)

)+ β3λV 2

(1 + β4λI)V4=V

0 − λM−1

∆x∆y

(α51L(V

0) + α52L(V

1) + α53L(V

2) + α54L(V

3))

+β4λV3

• We require from the 4- and 5-stage Runge-Kutta schemes that they are second

order accurate


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 using

quadratic basis-functions

• For the optimization process, we fix the Reynolds numbers Rex = Rey = 100,

the flow angle γflow = π/4 and the aspect ratio AR = 1.

• For steady flows we fix CFL∆t = 100, while for unsteady flows CFL∆t = 1.


• The optimization process has a big impact on the Runge-Kutta coefficients which

greatly differ per case.

• The Runge-Kutta coefficients are very different from commonly used schemes since

they are optimized for fast multigrid convergence and not for time accuracy.

• Finding a good balance between optimization and more generally applicable

algorithms is still an open question.

• In practice, the Runge-Kutta smoothers use local time stepping and this gives the

opportunity to apply locally the best smoother for the actual flow state.


Optimized coefficients for dRK5 and fRK5 smoothers for 3-level multigrid (steady flow).

dRK5 p = 1 fRK5 p = 1 dRK5 p = 2 fRK5 p = 2α21 0.05768995298 0.0578331573 0.04865009589 0.04877436325α31 - -0.0002051554736 - -0.0002188348438

α32 0.1405960888 0.1403808301 0.130316854 0.1300906122α41 - 0.0003953470071 - 2.608884832e-05

α42 - -0.001195029164 - 2.444376496e-05α43 0.267958213 0.2681810517 0.2729621396 0.2734805705

α51 - 0.0001441249202 - -0.001250385487α52 - -0.0002608610327 - -0.0007838720635

α53 - -0.0003368070181 - -0.0004890887712α54 0.5 0.8473374098 0.5 4.412139367

α61 - 0.4115573097 - 0.8097217358α62 - -0.003144851878 - 0.08435089009

α63 - -0.0001096455683 - -0.01986799007α64 - 0.001555741114 - 0.01359815476

α65 1.0 0.5901414466 1.0 0.1121972094

CFLτ 0.8 0.8 0.4 0.4

ρS 0.98812 0.98914 0.98974 0.9896

ρMG 0.89151 0.81762 0.90049 0.89903

ρEXI−MG 167.06 - 124.02 -


−10 −8 −6 −4 −2 0 2−5

−4

−3

−2

−1

0

1

2

3

4

5Stability domain of smoother

0.1

0.1

0.1

0.1

0.1

0.1

0.2

0.2

0.2

0.20.20.2

0.3

0.3

0.3

0.30.3

0.3

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.5

0.5

0.60.6

0.6

0.60.6

0.6

0.7

0.70.7

0.7

0.7

0.7

0.7

0.8

0.80.8

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 dRK5.

−10 −8 −6 −4 −2 0 2−5

−4

−3

−2

−1

0

1

2

3

4

5

Re(z)

Im(z

)

Spectrum of L and stability domain of smoother

lowhigh

(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.


−10 −8 −6 −4 −2 0 2−5

−4

−3

−2

−1

0

1

2

3

4

5Stability 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

0.30.3

0.4

0.4 0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.5

0.5

0.60.6

0.6

0.60.6

0.6

0.7

0.70.7

0.7

0.70.7

0.8

0.80.8

0.8

0.80.8

0.8

0.9

0.90.9

0.9

0.90.9

0.9

1

1 1

1

11

1

Re(z)

Im(z

)

(a) Stability domain fRK5.

−10 −8 −6 −4 −2 0 2−5

−4

−3

−2

−1

0

1

2

3

4

5

Re(z)

Im(z

)

Spectrum of L and stability domain of smoother

lowhigh

(b) Spectrum of Lh 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.


−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

lowhigh

(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

lowhigh

(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 three−level 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 three−level operator

(d) 3-level MG with fRK5.

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

equation using quadratic basis functions.


Testing multigrid performance

• In order to demonstrate the performance of the optimized algorithms we consider

the 2D advection-diffusion equation on a square.

• The exact steady state solution is

u(x1, x2) =1

2

(exp(a1/ν1) − exp(a1x1/ν1)

exp(a1/ν1) − 1+

exp(a2/ν2) − exp(a2x2/ν2)

exp(a2/ν2) − 1

).


• The advection-diffusion equation is discretized using the space-time discontinuous

Galerkin discretization with quadratic basis functions.

• In the discretization we use a Shishkin mesh which is suitable for dealing with

boundary layers.

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


• The parameters in the tests are:

I ∆t = 100, a =√

2, νx = νy = 0.01, N1 = N2 = 32.

I Flow angle γflow = π/4.

I Depending on the stability of the smoother, we use different CFLτ and V Nτ

numbers.

I The EXV scheme was used when Rei ≤ 1 and the EXI scheme was used

otherwise.

I For the multigrid computations, ν1 = ν2 = ν3 = ν4 = 1, νC = 4 and γ = 1.


Work units

L/L

0

500 1000 1500 2000 2500 300010-4

10-3

10-2

10-1

dRK5 coarse approx.fRK5 coarse approx.EXI/EXV coarse approx.dRK5 coarse exactfRK5 coarse exactEXI/EXV coarse exact

Convergence of three-level multigrid with different Runge-Kutta smoothers for second

order accurate space-time DG discretization.


Work units

L/L

0

5000 10000 15000 2000010-4

10-3

10-2

10-1 dRK5 coarse approx.fRK5 coarse approx.EXI/EXV coarse approx.dRK5 coarse exactfRK5 coarse exactEXI/EXV coarse exact

Convergence of three level multigrid with different Runge-Kutta smoothers for third

order accurate space-time DG discretization.


We can draw the following conclusions:

• In all cases using the optimized Runge-Kutta smoothers a big improvement is

obtained 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 from

22000 to 184.

• For linear basis functions the difference in convergence rate between Runge-Kutta

smoothers with only non-zero diagonal terms versus Runge-Kutta smoothers with

all coefficients non-zero is negligible.


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

• Using more Runge-Kutta coefficients enlarges the possibilities to optimize the

smoother, but the optimization process requires a significantly larger computing

time.

• In order to speed up the optimization process the coefficients of Runge-Kutta

schemes with only non-zero diagonal terms are used as initial values.


• 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 on

the 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 equation

directly to the Euler and Navier-Stokes equations.


Euler Equations

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

work per cycle =

gpbp, SG,

(gpbp + gp−1bp−1)(νp1 + νp

2) + gp−2bp−2νpC, pMG,

gpbp((ch + ch−1)(νh

1 + νh2 ) + ch−2νh

C

), hMG,

ch(gpbp + gp−1bp−1)(νp1 + νp

2)+

gp−2bp−2((ch + ch−1)(νh1 + νh

2 ) + ch−2νhC), 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, gp−1 = 4, gp−2 = 1, bp = 6, bp−1 = 4, bp−2 = 1, ch = 1,

ch−1 = 1/4 and ch−2 = 1/16.


p-levels h-levels CF Lτ ρ 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.


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

L 2 to

tal

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, α = 2, Ma = 0.5,

448 × 64 elements).


Conclusions

• A space-time DG discretization for nonconservative hyperbolic pde’s using the

DLM theory and the compressible Navier-Stokes has been developed.

• A new numerical flux for nonconservative hyperbolic pde’s has been developed,

which reduces to the HLLC flux for conservative pde’s.

• The effect of the choice of the path in phase space is in practice for nearly all cases

negligible.

• The algorithm has been successfully tested on a depth averaged two-phase flow

model and compressible flow simulations.


Conclusions

• A detailed three-level multigrid analysis has been conducted for h-, p- and

hp-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 significant

improvement in the convergence rate in actual computations using the h-multigrid

algorithm.


• For the Euler equations the optimized Runge-Kutta smoothers for h-, p- and

hp-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 for

the Euler equations.

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