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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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 2 / 33
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 3 / 33
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 4 / 33
SD Method for Triangles
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 5 / 33
SD Method for Triangles
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 6 / 33
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)
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 7 / 33
SD Method with Raviart-Thomas Elements
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 8 / 33
SD Method with Raviart-Thomas Elements
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 9 / 33
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 10 / 33
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 11 / 33
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]
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 12 / 33
LSA - Spatial Discretization
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]
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 13 / 33
LSA - Spatial Discretization
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 14 / 33
LSA - Spatial Discretization
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 15 / 33
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 16 / 33
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 17 / 33
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 18 / 33
Results - 2D Euler Equations
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 19 / 33
Results - 2D Euler Equations
Test case 1 - Free stream Mach number - 0.3, Angle of attack - 0 degree
Figure: Mach contours - RT2 scheme
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 20 / 33
Results - 2D Euler Equations
Test case 1 - Convergence of residual
0 5 10 15 20 25−14
−12
−10
−8
−6
−4
−2
0
Iterations
Log(
Res
)
RT1RT2RT3
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 21 / 33
Results - 2D Euler Equations
Test case 1 -Comparison of 4th order DG and RT3 schemes
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 22 / 33
Results - 2D Euler Equations
Test case 1 - Free stream Mach number - 0.3, Angle of attack - 0 degree
Figure: Mach contours - RT1 scheme
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 23 / 33
Results - 2D Euler Equations
Test case 1 - Free stream Mach number - 0.3, Angle of attack - 0 degree
Figure: Mach contours - RT2 scheme
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 24 / 33
Results - 2D Euler Equations
Test case 1 - Free stream Mach number - 0.3, Angle of attack - 0 degree
Figure: Mach contours - RT3 scheme
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 25 / 33
Results - 2D Euler Equations
Test case 2 - Free stream Mach number - 0.4, Angle of attack - 5 degree
Figure: Mach contours - RT2 scheme
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 26 / 33
Results - 2D Euler Equations
Test case 2 - Convergence of Residual
0 5 10 15 20 25−14
−12
−10
−8
−6
−4
−2
0
Iterations
Log(
Res
)
RT1RT2RT3
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 27 / 33
Results - 2D Euler Equations
Test case 2 -Comparison of 4th order DG and RT3 Schemes
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 28 / 33
Results - 2D Euler Equations
Test case 2 - Free stream Mach number - 0.4, Angle of attack - 5 degree
Figure: Mach contours - RT1 scheme
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 29 / 33
Results - 2D Euler Equations
Test case 2 - Free stream Mach number - 0.4, Angle of attack - 5 degree
Figure: Mach contours - RT2 scheme
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 30 / 33
Results - 2D Euler Equations
Test case 2 - Free stream Mach number - 0.4, Angle of attack - 5 degree
Figure: Mach contours - RT3 scheme
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 31 / 33
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 32 / 33
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 33 / 33
LSA - Spatial Discretization
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 34 / 33
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.
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 35 / 33
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
.
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 36 / 33
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),
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 37 / 33
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
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 38 / 33
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
)
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 39 / 33
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
)
Aravind Balan (AICES, RWTH Aachen) New Spectral Difference Jan 4, Orlando 40 / 33