A Stable Spectral Difference Method for Triangles Webseite...A Stable Spectral Difference Method for...

A Stable Spectral Difference Method for Triangles

Aravind Balan1, Georg May1, and Joachim Schoberl2

1AICES Graduate School, RWTH Aachen, Germany

2Institute for Analysis and Scientific Computing, Vienna Technical University, Austria

AIAA Aerospace Sciences MeetingJanuary 4, 2011Orlando, Florida

Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 1 / 33

Outline

1 Background and Motivation

2 Spectral Difference(SD) Method

3 SD Method with Raviart-Thomas Elements

4 Stability Analysis

5 Results


Background and Motivation

Spectral Difference (SD)→ high-order method for hyperbolic PDEs

A quadrature free (pre-integrated) nodal Discontinuous Galerkin scheme

Simple in formulation and implementation

Found linearly unstable for triangles for order of accuracy > 2

Found stable with flux interpolation on Raviart-Thomas elements


SD Method for Triangles

Hyperbolic conservation equation∂u(~x,t)∂t +∇ · ~f (u) = 0

Transformation from reference element Φ : ξ → x with J = ∂x/∂ξ(0,1)

(1,0)(0,0)x

y

x1

x3

x2

ξ

η

Φ :(ξ, η) (x,y)

Hyperbolic equation in reference domain∂u(ξ,t)∂t + 1

|J|∇ξ ·(|J |J−1 ~f (u)

)= 0

DefineSolution collocation nodes - ξj

Flux collocation nodes - ξj



Approximation of solution

uh (ξ) =∑Nm

j=1 uj lj (ξ) lj ∈ Pm

lj(ξk) = δjk ⇒ uj = uh

(ξj

)no. of degrees of freedom = Nm = (m+1)(m+2)

2

Approximation of flux~fh (ξ) =

∑Nm+1

j=1~fj lj (ξ) lj ∈ Pm+1

lj(ξk) = δjk ⇒ ~fj = ~fh

(ξj

)~fj =

{|J |J−1 ~f

(ξj

)ξj ∈ T

~fnum ξj ∈ ∂T~fnum · n = h → standard numerical flux

no. of degrees of freedom = 2Nm+1



Final form of the Spectral Difference scheme

du(i)j

dt+

1

|J (i)|

Nm+1∑k=1

∇ξ lk(ξj

)· ~f (i)k = 0, j = 1, ..., Nm

Linearly unstable for m ≥ 2 for triangles [Van den Abeele et al., 2008 ]

Note - Each of the flux vectors need not be in Pm+1 for the div to be in Pm

→ Raviart Thomas space→ Smallest space having div in Pm


SD Method with Raviart-Thomas Elements

DefineSolution points - ξj

Flux points - ξj , Directions - sj

Approximation of solutionuh (ξ) =

∑Nm

j=1 uj lj (ξ)

lj(ξk) = δjk ⇒ uj = uh

(ξj

)Approximation of flux function in Raviart-Thomas (RT ) space

~fh (ξ) =∑NRT

mj=1 fj ~ψj (ξ)

~ψj(ξk) · sk = δjk ⇒ fj = ~fh

(ξj

)· sj

fj =

{|J |J−1 ~f

(ξj

)· sj ξj ∈ T

h ξj ∈ ∂Th → standard numerical fluxNRTm = (m+ 1) (m+ 3)



For a degree m, the RT space is defined as

RTm = [Pm]2

+ (x, y)TPm.

For m = 1, the monomials which form a basis in the RT space(10

),

(x0

),

(y0

),

(01

),

(0x

),

(0y

),

(x2

yx

),

(xyy2

)Less number of flux degrees of freedom compared to standard SD

2Nm+1 −NRTm = m+ 3

Flux nodes distribution :

m+ 1 nodes on each edge and NRTm − 3(m+ 1) in the interior



Final form of the new Spectral Difference scheme

du(i)j

dt+

1

|J (i)|

Nrtm∑

k=1

f(i)k

(∇ξ · ~ψk

)(ξj

)= 0, j = 1, ..., Nm

Linearly stable for m = 1, 2, 3 in a simplified stability analysis

Numerical experiments prove the viability of the scheme


Linear Stability Analysis

Linear advection equation

∂u(~x, t)

∂t+∇ · ~f (u) = 0, ~f(u) = (u|c|cosθ, u|c|sinθ), θ ∈ [0,

π

2]

Consider Cartesian mesh with each element formed by fusing twotriangles

i

j

SD formulation, using upwind fluxes

∆tU (i,j) = −ν(AU (i,j) +BU (i−1,j) + CU (i,j−1)

),

Linear stability analysis (LSA)⇒ Fourier transformation : u→ u


Linear Stability Analysis

SD discretization of the Fourier mode uei(kxx+kyy)

du

dt=

ν

∆tZu Z = −

(A+Be−iσ + Ceiκ

)(σ, κ) = (kxh, kyh)

Full stability = Stability of spacial discretization + time discretization

Stability of spacial discretization→ eigensystem of Z

Stable flux nodes→ Re(λ(Z)) ≤ 0

Optimal flux nodes→Max(|λ(Z)|) (Spectral Radius) is minimum

Stability is independent of the position of solution nodes

Stability is independent of the position of flux nodes on the edges


LSA - Spatial Discretization

RT1 → 1 interior flux point at centroid - stable

RT2 → 3 interior points each with two ortho directions form 6 flux nodes

3 interior points are varied as

ξi = ξc + α(ξei − ξc), i = 1, 2, 3 α ∈ [0, 1]

Stable→ 0.5 ≤ α ≤ 0.521, considering θ ∈ [0, π2]

Stable and optimal→ α = 0.5 [higher order quadrature nodes]



RT3 → 6 interior points each with two ortho directions form 12 flux nodes6 interior points are varied as

ξi = ξc + α(ξei − ξc), i = 1, 2, 3 α ∈ [0, 1]

ξi = ξc + β(ξei − ξc), i = 4, 5, 6 β ∈ [0, 1]

Stable and optimal→ α = 0.725 β = 0.676 [higher order quadraturenodes]



Stability and optimality for RT3

!

"

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Stable Region



Stable and optimal flux points

RT10.4 0.2 0 0.2 0.4 0.6 0.8 1

0.2

0

0.2

0.4

0.6

0.8

1

x

y

RT20.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

y

RT3

0.4 0.2 0 0.2 0.4 0.6 0.8 1 1.2

0.2

0

0.2

0.4

0.6

0.8

1

x

y


LSA - Full Discretization

Full discretization→ un+1 = Gun

L2 stability→ ρ(G) ≤ 1→ Get allowable CFL number (= |c|∆th )

Max CFL number for Shu-RK3 time discretization

0 5 10 15 20 25 30 35 40 450.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

!

CFL

num

ber

RT1RT2RT3


Results - 2D Linear Advection Equation

Linear advection equation

∂u(~x, t)

∂t+∇ · ~f (u) = 0, ~f(u) = (cxu, cyu)T

Convergence study

1 1.2 1.4 1.6 1.8 2−8

−7

−6

−5

−4

−3

−2

−1

Log(N)

Log(

L ∞ E

rror

)

RT1RT2RT3


Results - 2D Euler Equations

Euler equations in 2D

∂f(u)

∂x+∂g(u)

∂y= 0

u =

ρρuρvE

f =

ρu

ρu2 + pρuv

u (E + p)

g =

ρvρuv

ρv2 + pv (E + p)

,NACA0012 - 1440 mesh elements

X

Y

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5



Relaxation Technique

Backward Euler / Damped Newton

(I −∆t

dR(Un)

dU

)∆Un = ∆tR(Un),

∆t→∞⇒ Newton iteration (Quadratic convergence)

Preconditioning - Incomplete LU factorization

Linear system solver - Restarted GMRES algorithm



Test case 1 - Free stream Mach number - 0.3, Angle of attack - 0 degree

Figure: Mach contours - RT2 scheme



Test case 1 - Convergence of residual

0 5 10 15 20 25−14

−12

−10

−8

−6

−4

−2

0

Iterations

Log(

Res

)

RT1RT2RT3



Test case 1 -Comparison of 4th order DG and RT3 schemes



















Test case 2 - Convergence of Residual

0 5 10 15 20 25−14

−12

−10

−8

−6

−4

−2

0

Iterations

Log(

Res

)

RT1RT2RT3



Test case 2 -Comparison of 4th order DG and RT3 Schemes














Summary and Outlook

Difference between SD and new SD

New SD formulation is found to be linearly stable

Numerical results show the viability

Needs to be extended to solve NS equations, also to simulate transonicflows


Acknowledgement

Financial support from the Deutsche Forschungsgemeinschaft(German Research Association) through grant GSC 111, and by

the Air Force Office of Scientific Research, Air Force MaterielCommand, USAF, under grant number FA8655-08-1-3060, is

gratefully acknowledged



Stability and optimality for RT2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

!

max

",#

,$ R

e(%)

0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 x 10−3

!

max

",#

Re($)

00.2

0.40.6

0.81

0

20

40

600

20

40

60

80

100

max

,

(Z)

10

20

30

40

50

60

70

80


Runge-Kutta Time stepping

The solution at (n+ 1)-th iteration, Un+1, is obtained from Un as

w(0) = Un,

w(k) =

k−1∑l=0

αklw(l) + ∆tβklR

(l) k = 1, ..., p,

Un+1 = w(p),

For stability analysis

w(0) = Un,

w(k) =

k−1∑l=0

αklw(l) + νβklZw

(l) k = 1, ..., p, (1)

Un+1 = w(p).

If G(k) is the amplification matrix in the k-th intermediate step, then oneobtains

G(0) = I, G(k) =

k−1∑l=0

(αklI + νβklZ)G(l) k = 1, ..., p.


Runge-Kutta Time Stepping

Shu-RK3

α =

134

14

13 0 2

3

, β =

10 1

40 0 2

3

.5 stage 4th order SSP

α =

1

0.4443704940 0.55562950590.6201018513 0 0.37989814860.1780799541 0 0 0.82192004580.0068332588 0 0.5172316720 0.1275983113 0.3483367577

,

β =

0.3917522270

0 0.36841059260 0 0.25189177420 0 0 0.54497475020 0 0 0.0846041633 0.2260074831

.


Backward-Euler Time Stepping

Backward Euler

Un+1 − Un

∆t= R(Un+1),

Taylor series for R

R(Un+1) = R(Un) +dR(Un)

dt∆t+ ....

Rewrite

dR(Un)

dt∆t =

dR(Un)

dU

dU

dt∆t ' dR(Un)

dU(Un+1 − Un).

If (Un+1 − Un) is denoted as ∆Un, the implicit scheme is(I −∆t

dR(Un)

dU

)∆Un = ∆tR(Un),


Stability Analysis

Numerical schemes should posess non-linear stability properties

Eg. Total Variation Diminishing (TVD)

TV (un) =

NT∑i=1

|uni+1 − uni |,

TV(un+1

)≤ TV (un) .

TVD property→ convergence

A conservative numerical scheme can be made to satisfy TVD by usinglimiters

Linearly unstable→ limiter will act on smooth regions→ affect order ofaccuracy


Finding Transfer Matrix

Solution interpolation

uh (ξ) =

Nm∑j=1

uj lj (ξ) uh

(ξj

)= uj

Solution at flux nodes

uh

(ξk

)=

Nm∑j=1

uj lj

(ξk

)Dubiner basis

χ (ξ) =

Nm∑j=1

χj lj (ξ) χj = χ(ξj

)Dubiner basis at flux nodes

χ(ξk

)=

Nm∑j=1

χj lj

(ξk

)


Finding Differentiation Matrix

The monomials in the RT space at solution nodes

~φn

(ξj

)=

NRTm∑k=1

an,k ~ψk

(ξj

)an,k = ~φn

(ξk

)· sk

Its divergence

∇ξ · ~φn(ξj

)=

NRTm∑k=1

an,k

(∇ξ · ~ψk

)(ξj

)


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