Thesis Defence (pdf) - Department of Aerospace Engineering ...

Middle East Technical University
Aerospace Engineering Department
Effects of the Jacobian Evaluation on
Direct Solutions of the Euler Equations
by Ömer Onur
Supervisor: Assoc. Prof. Dr. Sinan Eyi

Outline
Introduction
Objectives
Flow Analysis
Accuracy Analysis & Results
Performance Analysis & Results
Conclusion
Recommendations
Ömer Onur - AE500 Presentation
2

Introduction
Ömer Onur - AE500 Presentation
Steady flow computations can be realized either by
iterative means, means,
or using direct methods. methods
Although iterative solvers make an unsteady flow
analysis with an advance in time, time,
with direct methods
a steady flow analysis is possible.
Low memory requirements made the iterative
methods popular untill the development of recent
advanced computers.
Solving the whole domain at once makes direct
solvers stable, and faster convergence is possible
due to small number of iterations. iterations
In CFD applications like design optimization, optimization,
and
flutter analysis, analysis,
direct methods are preferable.
3

Introduction (Cont.)
Ömer Onur - AE500 Presentation
Direct solvers require the calculation of Jacobian
matrix.
Derivation of analytical nalytical Jacobians becomes more
difficult as the discretization of governing equations
become more complex.
The best alternative is to compute the Jacobians
numerically as accurate as possible.
4

Objectives
Ömer Onur - AE500 Presentation
To develop a direct flow solver code
To compare the accuracy of numerical and analytical
Jacobians used in direct flow solvers
To investigate their effects on the performance
(convergence convergence and CPU time) of direct flow solvers
To improve the efficiency of Jacobian matrix solution
5

Flow Analysis
Ömer Onur - AE500 Presentation
Flow code is a 2-D planar/axisymmetric Euler solver:
– 1st/2nd order finite-volume discretizations
– Steger-Warming, Van Leer, Roe upwind flux splitting
schemes
– Newton's method
– both numerical and analytical Jacobian matrices
– UMFPACK sparse matrix solver package
Different geometries, geometries,
grid sizes and BCs are
considered.
6

Flow Analysis (Cont.)
Ömer Onur - AE500 Presentation
Steady, 2-D planar/axisymmetric Euler equations in
generalized coordinates:
∂F(W ˆ ˆ ) ∂G(W
ˆ ˆ )
+ + σ H(W ˆ ˆ ) = 0
∂ξ ∂η
ˆF = J
⎡ ρU ⎤
⎢
ρuU + ξ p
⎥
⎢ ρvU + ξ p ⎥
−1
⎢ x ⎥
y
⎢ ⎥
⎣( ρet
+ p)U ⎦
ˆG = J
⎡ ρV ⎤
⎢
ρuV + η p
⎥
−1
⎢ x ⎥
⎢ ρvV + η p ⎥
y
⎢ ⎥
⎣( ρet
+ p)V ⎦
where
ˆH = J
−1
ˆW = J
−1
⎡ρ⎤ ⎢
ρu
⎥
⎢ ⎥
⎢ρv⎥ ⎢ ⎥
ρe
⎣ t ⎦
⎡ρv⎤ ⎢
1 ρuv
⎥
⎢ ⎥
2
y ⎢ρv⎥ ⎢ ⎥
⎢( ρe
+ p)v⎥
⎣ t ⎦
7

Flow Analysis (Cont.)
Newton’s method:
R(W ˆ ˆ ) = 0
n
ˆR ˆ n ˆ n
⎛ ∂ ⎞
⎜ ˆW
⎟
⎝∂⎠ (Discretized Residual)
⋅ ∆W
= −R(W
)
[Jacobian Matrix]
Ömer Onur - AE500 Presentation
ˆ +
W = Wˆ + ∆Wˆ
n 1 n n
8

Flow Analysis (Cont.)
Analytical
Ömer Onur - AE500 Presentation
Analytical Jacobians:
Jacobians
– Derivation by hand / symbolic manipulators for simple
flow models
– Difficult derivation for complex flow models
Numerical Jacobians:
Jacobians
∂Rˆ
i
∂ ˆW
R(Wˆ ˆ
i + ε ⋅e j ) − R(W) i
=
ε
j
– Good choice of finite-difference perturbation magnitude ε
– Usage of higher computer precision
9

Accuracy Analysis
Ömer Onur - AE500 Presentation
Error analysis:
– Condition error [loss of numerical precision]
– Truncation error [neglected terms in the Taylor series]
– Total error
2
2⋅ER ∂ f ( ξ ) ε
E TOTAL( ε ) = E C( ε ) + E T ( ε ) = + ⋅ 2
ε ∂x
2
ε> & ε >> E T>>
ε opt
E TOTALmin
10

Accuracy Analysis (Cont ( Cont.) .)
Optimum perturbation magnitude analysis:
∂E ( ε ) 2⋅E 1
2
TOTAL R ∂ f( ξ )
=− + ⋅ = 0
∂ε ε
2 2 ∂x
2
Ömer Onur - AE500 Presentation
= 2 ⋅
∂
ER
f( ξ )
2
∂x
OPT 2
– E R can be taken as ε M or simply found by subtraction of
double and single precision calculations of flux values.
– Second derivative of flux can be calculated using a
forward/backward finite difference method with double
precision or taken as equal to 1.
– Optimum perturbation magnitude is also found by trialerror
ε
11

Accuracy Results
Ömer Onur - AE500 Presentation
Error in numerical flux Jacobians are analyzed for
– 15º ramp geometry with 33x25 grid
– M=2.0 free-stream flow
– inlet-outlet, symmetry and wall BCs
Jacobians are calculated using
– already converged solution
– 1st order S-W flux splitting
Different perturbation magnitude values are
investigated for both single and double precision.
Optimum perturbation magnitude analysis is
performed for single precision.
12

TotalError between ResidualJacobians (∂ R/∂ W)
10 4
10 3
10 2
10 1
10 0
10 -1
10 -2
10 -3
10 -4
10 1 10 -5
Accuracy Results (Cont.)
Effect of Control on Total Errors for Residual Jacobians
[Single Precision, Forward Differencing]
10 -1
ε =7×10 -4
10 -3
10 -5
Perturbation Magnitude (ε)
w/o Cont. (Max. E)
wCont.(Max.E)
w/o Cont. (Avg. E)
w Cont. (Avg. E)
10 -7
10 -9
TotalError between ResidualJacobians (∂ R/∂ W)
10 4
10 3
10 2
10 1
10 0
10 -1
10 -2
10 -3
10 -4
10 1 10 -5
10 -1
Ömer Onur - AE500 Presentation
Effect of Control on Total Errors for Residual Jacobians
[Single Precision,Backward Differencing]
ε =7×10 -4
10 -3
10 -5
Perturbation Magnitude (ε)
w/o Cont. (Max. E)
wCont.(Max.E)
w/o Cont. (Avg. E)
w Cont. (Avg. E)
10 -7
13
10 -9

TotalMax. Error between Jacobians
10 4
10 3
10 2
10 1
10 0
10 -1
10 -2
10 1 10 -3
Accuracy Results (Cont.)
Change of Total Maximum Error with Perturbation Magnitude
[Single Precision, Forward Differencing]
10 -1
ε =7×10 -4
10 -3
10 -5
Perturbation Magnitude (ε)
10 -7
∂ F + / ∂ W
∂ F - / ∂ W
∂ G + / ∂ W
∂ G - / ∂ W
∂ R pl / ∂ W
10 -9
Total Avg. Error between Jacobians
10 3
10 2
10 1
10 0
10 -1
10 -2
10 -3
10 -4
10 1 10 -5
10 -1
Ömer Onur - AE500 Presentation
Change of Total Average Error with Perturbation Magnitude
[Single Precision, Forward Differencing]
ε =7×10 -4
10 -3
10 -5
Perturbation Magnitude (ε)
10 -7
∂ F + / ∂ W
∂ F - / ∂ W
∂ G + / ∂ W
∂ G - / ∂ W
∂ R pl / ∂ W
14
10 -9

TotalError between ResidualJacobians (∂ R/∂ W)
10 4
10 3
10 2
10 1
10 0
10 -1
10 -2
10 -3
10 -4
10 1 10 -5
Accuracy Results (Cont.)
Effect of Axisymmetry on Total Errors for Residual Jacobians
[Single Precision, Forward Differencing]
10 -1
ε =7×10 -4
10 -3
10 -5
Perturbation Magnitude (ε)
Planar (Avg. E)
Axis ym. (Avg. E)
Planar (Max. E)
Axisym. (Max. E)
10 -7
10 -9
Ömer Onur - AE500 Presentation
Optimum Perturbation Magnitude (εopt) Analysis
for Single precision
Trial-Error Procedure Optimization Method
Flux εopt (max. error) εopt (avg. error) εopt (avg.) εopt ( εM)
F +
9.8 . 10 -4
6.4 . 10 -4
7.7 . 10 -4
F -
G
- - -
+
1.5 . 10 -5
6.4 . 10 -4
7.9 . 10 -4
G -
1.5 . 10 -5
6.4 . 10 -4
7.8 . 10 -4
3.5.10 -4
For single precision;
ε opt ≈ 7x10 -4
15

TotalError between ResidualJacobians (∂ R/∂ W)
10 3
10 1
10 -1
10 -3
10 -5
10 -7
10 -2 10 -9
Accuracy Results (Cont.)
Effect of Control on Total Errors for Residual Jacobians
[Double Precision, Forward Differencing]
10 -4
10 -6
ε =4×10 -8
10 -8
Perturbation Magnitude (ε)
w/o Cont. (Max. E)
wCont.(Max.E)
w/o Cont. (Avg. E)
w Cont. (Avg. E)
10 -10
10 -12
TotalError between ResidualJacobians (∂ R/∂ W)
10 3
10 1
10 -1
10 -3
10 -5
10 -7
10 -2 10 -9
10 -4
Ömer Onur - AE500 Presentation
Effect of Control on Total Errors for Residual Jacobians
[Double Precision, Backward Differencing]
10 -6
ε =4×10 -8
10 -8
Perturbation Magnitude (ε)
w/o Cont. (Max. E)
wCont.(Max.E)
w/o Cont. (Avg. E)
w Cont. (Avg. E)
10 -10
16
10 -12

TotalMax. Error between Jacobians
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
10 -8
10 -2 10 -9
Accuracy Results (Cont.)
Change of Total Maximum Error with Perturbation Magnitude
[Double Precision, Forward Differencing]
10 -4
10 -6
ε =4×10 -8
10 -8
Perturbation Magnitude (ε)
10 -10
∂ F + / ∂ W
∂ F - / ∂ W
∂ G + / ∂ W
∂ G - / ∂ W
∂ R pl / ∂ W
10 -12
TotalAvg. Error between Jacobians
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
10 -8
10 -9
10 -2 10 -10
10 -4
Ömer Onur - AE500 Presentation
Change of Total Average Error with Perturbation Magnitude
[Double Precision, Forward Differencing]
10 -6
ε =4×10 -8
10 -8
Perturbation Magnitude (ε)
10 -10
∂ F + / ∂ W
∂ F - / ∂ W
∂ G + / ∂ W
∂ G - / ∂ W
∂ R pl / ∂ W
17
10 -12

TotalError between ResidualJacobians (∂ R/∂ W)
10 0
10 -1
10 -2
10 -3
10 -4
10 -5
10 -6
10 -7
10 -8
10 -2 10 -9
Accuracy Results (Cont.)
Effect of Axisymmetry on Total Errors for Residual Jacobians
[Double Precision, Forward Differencing]
10 -4
10 -6
ε =4×10 -8
10 -8
Perturbation Magnitude (ε)
Planar (Avg. E)
Axis ym. (Avg. E)
Planar (Max. E)
Axisym. (Max. E)
10 -10
10 -12
Ömer Onur - AE500 Presentation
Optimum Perturbation Magnitude (εopt) Analysis
for Double precision
Trial-Error Procedure Optimization Method
Flux εopt (max. error) εopt (avg. error) εopt
F +
4 . 10 -8
4 . 10 -8
F -
- -
G +
4 . 10 -8
4 . 10 -8
G -
4 . 10 -8
4 . 10 -8
For double precision;
ε opt ≈ 4x10 -8
1.5 . 10 -8
18

Performance Analysis
Ömer Onur - AE500 Presentation
Jacobian Matrix structure:
– a huge square matrix even for a simple geometry
– composed of either analytical or numerical Jacobians
– most of the elements are zero
Matrix solver:
– Direct full matrix solvers have very high storage and
memory costs.
– Storage of only non-zero elements improve the
efficiency greatly.
Sparse Matrix Solvers
19

Ömer Onur - AE500 Presentation
Performance Analysis (Cont.)
Matrix Solution Strategies:
– Frozen Jacobian [using same Jacobian matrix in
subsequent iterations]
– Good initial guess [a time-like term addition to the Jacobian
matrix diagonal]
0
n
ˆ
R(W ˆ )
⎛ 1 ∂ R ⎞
n 0
n n
2
⎜ [] I + ∆Wˆ
= −R(W
ˆ ) ∆t = ∆t
∆t ∂Wˆ
⎟
ˆ n
R(W )
⎝ ⎠
2
Δt>>
Original Newton’s method
20

Performance Results
Ömer Onur - AE500 Presentation
Test Case Results:
Test Case Results:
– Convergence and CPU time performance of the direct
solver is analyzed for
15º ramp geometry with 33x25 grid
M=2.0 free-stream flow
inlet-outlet, symmetry and wall BCs
– Jacobians are calculated using
1st order S-W flux splitting
– Calculations are realized with forward and backward
differencing in single and double precision.
– Effect of perturbation magnitude ε is investigated
– Effect of Jacobian freezing and time-like diagonal term
addition are analyzed.
21

Performance Results (Cont Cont.) .)
33x25 Grid for Ramp geometry
Ömer Onur - AE500 Presentation
– 34x26 cells
– 4 flow variables
– total 3536
variables
– Jacobian matrix
has 35362 ≈ 12.5
million elements
Solver Storage CPU time
Full Matrix
LU
240 MB 26 h
Sparse
Matrix
1.2 MB 20 s
22

Performance Results (Cont Cont.) .)
Mach Contour for Ramp geometry
[2-D planar, Double Precision, 1st order S-W]
1.25731
1.95049
1.35633
1.45536
1.50487
1.75244
1.55439
1.65341
1.75244
1.55439
1.6039
1.45536
1.70292
1.35633
1.80195
Ömer Onur - AE500 Presentation
Mach Contour for Ramp geometry
[2-D axisymmetric, Double Precision, 1st order S-W]
1.65341
1.80195
1.90097
1.85146
1.98426
1.93113
1.95049
1.96313
1.98426
1.95049
23
1.97467

Max. Density Residual (R 1 )
10 0
10 -5
10 -10
10 -15
Performance Results (Cont Cont.) .)
Effect of ∆t onConvergenceHistory
[2-D planar, Double Precision]
No ∆t
∆t rm =1.5
∆t rm =5.
Full ∆t
AnalyticalJac.
No frz.
10
0 10 20
#ofiterations
30 40
-20
∆t
CPU Time
(s)
No ∆t NaN
∆trm= 1.5 13.32
∆trm= 5. 22.65
Full ∆t 257.27
Max. Density Res idual (R 1 )
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
10 -14
10 -16
No ∆t
∆t rm =1.5
∆t rm =5.
Full ∆t
Ömer Onur - AE500 Presentation
Effect of ∆t onConvergenceHistory
[2-D axisymmetric, Double Precision]
AnalyticalJac.
No frz.
10
0 5 10
#ofiterations
15 20
-18
∆t
CPU Time
(s)
No ∆t 9.79
∆trm= 1.5 9.84
∆trm= 5. 12.28
Full ∆t 22.59
24

Max. Density Res idual (R 1 )
10 1
10 -1
10 -3
10 -5
10 -7
10 -9
10 -11
10 -13
10 -15
Performance Results (Cont Cont.) .)
EffectOfFreezingonConvergenceHistory
[2-D planar, Double Precision]
No frz.
R frz =1×10 -1
R frz =1×10 -2
R frz =1×10 -4
Analytica l J ac.
∆t rm =1.5
10
0 5 10
#ofiterations
15 20
-17
Tolerance
CPU Time
(s)
No freeze 13.32
Rfrz= 1x10 -1
12.85
Rfrz= 1x10 -2
12.20
Rfrz= 1x10 -4
12.51
Max. Density Residual (R 1 )
10 1
10 -1
10 -3
10 -5
10 -7
10 -9
10 -11
10 -13
10 -15
No frz.
R frz =1×10 -1
R frz =1×10 -2
R frz =1×10 -4
Ömer Onur - AE500 Presentation
Effect Of Freezing on Convergence History
[2-D axisymmetric, Double Precision]
Analytica l J a c.
No ∆t
10
0 5 10
#ofiterations
15 20
-17
Tolerance
CPU Time
(s)
No freeze 9.79
Rfrz= 1x10 -1
10.32
Rfrz= 1x10 -2
8.15
Rfrz= 1x10 -4
8.23
25

Max. Density Residual (R 1 )
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
10 -14
Performance Results (Cont Cont.) .)
Convergence History
[2-D planar, Double Precision, Forward Differencing]
An. J a c.
ε =4×10 -4
ε =4×10 -8
ε =4×10 -12
10
0 5 10
#ofiterations
15
-16
Jacobian
CPU Time
(s)
Analytic 12.69
ε = 4x10 -4
22.05
ε = 4x10 -8
16.27
ε = 4x10 -12
19.11
∆t rm =1.5
R frz =10 -4
Max. Density Residual (R 1 )
10 1
10 -1
10 -3
10 -5
10 -7
10 -9
10 -11
10 -13
10 -15
An. J ac.
ε =4×10 -4
ε =4×10 -8
ε =4×10 -12
Ömer Onur - AE500 Presentation
Convergence History
[2-D a xis ymmetric, Double P re cis ion, Forward Diffe rencing]
∆t rm =0.
R frz =10 -2
10
0 2 4 6 8 10
#ofiterations
-17
Jacobian
CPU Time
(s)
Analytic 8.25
ε = 4x10 -4
10.06
ε = 4x10 -8
9.72
ε = 4x10 -12
10.08
26

Performance Results (Cont Cont.) .)
Ömer Onur - AE500 Presentation
Different Flux Splitting Schemes:
Different Flux Splitting Schemes:
– Convergence and CPU time performance of the direct
solver is analyzed for the test case using
1st order Van Leer/Roe flux splitting
– Jacobians are calculated using
1st order Van Leer/Roe flux splitting numerically
1st order S-W flux splitting analytically
– Optimum perturbation magnitude ε opt is used in the
calculations.
– Calculations are realized with forward and backward
differencing in single and double precision.
– Benefits of using the same flux calculation scheme for
both Jacobian and residual calculation are analyzed.
27

Performance Results (Cont Cont.) .)
Mach Contour for Ramp geometry
[2-D planar, Double Precision, 1st order VL]
1.25731
1.95049
1.40585
1.75244
1.55439
1.65341
1.6039
1.50487
1.70292
1.80195
1.35633
Ömer Onur - AE500 Presentation
Mach Contour for Ramp geometry
[2-D axisymmetric, Double Precision, 1st order VL]
1.65341
1.90097
1.85146
1.98426
1.93113
1.95049
1.96313
1.97467
1.93113
28
1.95049

Performance Results (Cont Cont.) .)
Mach Contour for Ramp geometry
[2-D planar, Double Precision, 1st order Roe]
1.25731
1.30682
1.95049
1.45536
1.75244
1.55439
1.65341
1.6039
1.80195
1.45536
1.70292
1.35633
Ömer Onur - AE500 Presentation
Mach Contour for Ramp geometry
[2-D axisymmetric, Double Precision, 1st order Roe]
1.65341
1.93113
1.85146
1.98426
1.90097
1.93113
1.96313
1.96313
1.97467
1.96313
29

Max. Density Residual (R 1 )
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
10 -14
Performance Results (Cont Cont.) .)
Convergence History
[2-D planar, Double Precision, S-W An. Jac.]
Steger-Warming
Va n Le e r
Roe
∆t rm =1.5
R frz =10 -4
10
0 50 100 150
#ofiterations
-16
FS
CPU Time
(s)
SW 12.69
VL 26.51
Roe 99.62
Max. Density Residual (R 1 )
10 2
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
10 -14
Ömer Onur - AE500 Presentation
Convergence History
[2-D planar, Double Precision, Num. Jac.]
Steger-Warming
Van Leer
Roe
∆t rm =1.5(SW)
∆t rm =2.(VL)
∆t rm =3.(Roe)
R frz =10 -4
ε =4×10 -8
10
0 5 10
#ofiterations
15 20
-16
FS
CPU Time
(s)
SW 16.27
VL 28.44
Roe 29.40
30

Max. Density Residual (R 1 )
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
10 -14
Performance Results (Cont Cont.) .)
Convergence History
[2-D axisymmetric, Double Precision, S-W An. Jac.]
Steger-Warming
Van Leer
Roe
10
0 50
#ofiterations
100
-16
FS
CPU Time
(s)
SW 8.25
VL 18.33
Roe 128.81
∆t rm =0.
R frz =10 -2
Max. Density Residual (R 1 )
10 2
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
10 -14
Ömer Onur - AE500 Presentation
Convergence History
[2-D axisymmetric, Double Precision, Num. Jac.]
Steger-Warming
Van Leer
Roe
10
0 5 10
#ofiterations
-16
FS
CPU Time
(s)
SW 9.72
VL 11.53
Roe 14.99
∆t rm =0.(SW,VL)
∆t rm =1.5(Roe)
R frz =10 -2
ε =4×10 -8
31

Performance Results (Cont Cont.) .)
Ömer Onur - AE500 Presentation
Higher-Order Discretizations:
Higher-Order Discretizations:
– Convergence and CPU time performance of the direct
solver is analyzed for the test case using
2nd order S-W/Van Leer/Roe flux splitting
Van Albada limiter
– Jacobians are calculated using
2nd order S-W/Van Leer/Roe flux splitting numerically
– Optimum perturbation magnitude ε opt is used in the
calculations.
– Calculations are realized with forward differencing in
double precision.
32

Performance Results (Cont Cont.) .)
Mach Contour for Ramp geometry
[2-D planar, Double Precision, 2nd order S-W]
1.25731
1.95049
1.30682
1.80195
1.40585
1.80195
1.50487
1.70292
1.85146
1.25731
1.90097
1.40585
Ömer Onur - AE500 Presentation
Mach Contour for Ramp geometry
[2-D axisymmetric, Double Precision, 2nd order S-W]
1.6039
1.75244
1.75244
1.85146
1.96313
1.98426
1.90097
1.93113
1.95049
33

Performance Results (Cont Cont.) .)
Mach Contour for Ramp geometry
[2-D planar, Double Precision, 2nd order VL]
1.95049
1.25731
1.35633
1.50487
1.75244
1.80195
1.45536
1.65341
1.85146
1.25731
1.45536
1.90097
Ömer Onur - AE500 Presentation
Mach Contour for Ramp geometry
[2-D axisymmetric, Double Precision, 2nd order VL]
1.65341
1.75244
1.95049
1.90097
1.85146
1.93113
1.98426
1.93113
34
1.95049
1.93113

Performance Results (Cont Cont.) .)
Mach Contour for Ramp geometry
[2-D planar, Double Precision, 2nd order Roe]
1.25731
1.95049
1.30682
1.75244
1.40585
1.65341
1.55439
1.85146
1.80195
1.25731
1.40585
Ömer Onur - AE500 Presentation
Mach Contour for Ramp geometry
[2-D axisymmetric, Double Precision, 2nd order Roe]
1.65341
1.75244
1.96313
1.85146
1.90097
1.98426
1.93113
1.90097
35
1.95049

Max. Density Residual (R 1 )
10 2
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
10 -14
Performance Results (Cont Cont.) .)
Convergence History
[2-D planar, Double Precision, Num. Jac., Van Albada]
Steger-Warming
Van Leer
Roe
Full ∆t
No Freezing
ε =4×10 -8
10
0 25 50
#ofiterations
75 100
-16
FS
CPU Time
(s)
SW 194.9
VL 334.67
Roe 335.28
Max. Density Residual (R 1 )
10 2
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
10 -14
Ömer Onur - AE500 Presentation
Convergence History
[2-D axisymmetric, Double Precision, Num. Jac., Van Albada]
Steger-Warming
Van Leer
Roe
Full ∆t
No Freezing
ε =4×10 -8
10
0 25 50
#ofiterations
75 100
-16
FS
CPU Time
(s)
SW Limit cycle
VL Limit cycle
Roe Limit cycle
36

Performance Results (Cont Cont.) .)
Ömer Onur - AE500 Presentation
Different Geometry and Flow Conditions:
Different Geometry and Flow Conditions:
– Convergence and CPU time performance of the direct
solver is analyzed for
bump geometry with 65x17 grid
M=2.0 and M=0.5 free-stream flows
inlet-outlet, symmetry and wall BCs
2nd order Roe flux splitting with Van Albada limiter
– Jacobians are calculated using
2nd order Roe flux splitting numerically
– Optimum perturbation magnitude ε opt is used in the
calculations.
– Calculations are realized with forward differencing in
double precision.
37

Performance Results (Cont Cont.) .)
65x17GridforBumpgeometry
Ömer Onur - AE500 Presentation
38

Performance Results (Cont Cont.) .)
Ömer Onur - AE500 Presentation
39

Max. Density Res idual (R 1 )
10 2
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
Performance Results (Cont Cont.) .)
Convergence History
[2-D planar, Double Precision, Num. Jac., Van Albada]
Roe
Full ∆t
No Freezing
ε =4×10 -8
10
0 100 200 300
#ofiterations
-14
Max. Density Res idual (R 1 )
10 2
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
Ömer Onur - AE500 Presentation
Convergence History
[2-D axisymmetric, Double Precision, Num. Jac., Van Albada]
Roe
Full ∆t
No Freezing
ε =4×10 -8
10
0 50 100 150
#ofiterations
200 250 300
-14
40

Performance Results (Cont Cont.) .)
Ömer Onur - AE500 Presentation
41

Max. Density Res idual (R 1 )
10 2
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
Performance Results (Cont Cont.) .)
Convergence History
[Subsonic 2-D planar, Double Precision, Num. Jac., Van Albada]
Roe
Full ∆t
No Freezing
ε =4×10 -8
10
0 100 200 300 400 500
#ofiterations
-14
Max. Density Res idual (R 1 )
Ömer Onur - AE500 Presentation
Convergence History
[Subsonic 2-D axisymmetric, Double Precision, Num. Jac., Van Albada]
10 2
10 0
10 -2
10 -4
10 -6
10 -8
10 -10
10 -12
Roe
Full ∆t
No Freezing
ε =4×10 -8
10
0 100 200 300 400 500
#ofiterations
-14
42

Conclusion
Ömer Onur - AE500 Presentation
A direct flow solver code is developed.
Accuracy of numerical Jacobians used in the solver is
analyzed.
– A control mechanism is required in 1st order SW
Jacobian calculation.
– Forward/backward differencing does not differ so much.
– Double precision improves accuracy significantly.
– Optimum perturbation magnitude is found by an
optimization method.
43

Conclusion (Cont ( Cont.) .)
Ömer Onur - AE500 Presentation
The effects of the accuracy of numerical Jacobians on
the performance (convergence convergence and CPU time) of
direct flow solvers is investigated.
– Double precision improves convergence limits
significantly.
– Optimum perturbation magnitude gives the same
convergence with the analytical method.
– Calculation of fluxes with perturbation only for related
cells reduced the execution time greatly.
44

Conclusion (Cont ( Cont.) .)
Ömer Onur - AE500 Presentation
The efficiency of Jacobian matrix solution is improved
by some strategies.
– A sparse matrix solver having low storage and memory
requiremets is used.
– A time-like term addition to matrix diagonal stabilizes
the solver in case of poor initial conditions. Removing
this modification at the right time makes the
convergence very fast.
– Jacobian matrix freezing at the right time decreases the
execution time.
45

Conclusion (Cont ( Cont.) .)
Ömer Onur - AE500 Presentation
Using the same flux calculation scheme for both
Jacobian and residual calculation maintains faster
convergence.
According to the choice of the flux limiter, higherorder
schemes may have convergence problems.
Unsteady results like limit-cycle is observed in higherorder
schemes.
46

Recommendations
Ömer Onur - AE500 Presentation
The control mechanism used in 1st order SW
Jacobian calculation can be improved to give the best
accuracy for all perturbation magnitude values.
More advanced strategies can be considered to
improve the Jacobian matrix solution.
Higher-order schemes, and especially the choice and
application of limiter can be analyzed deeply to obtain
fully converged solution.
47