1. Errors in Numerical Solutions of Shock Physics Problems
Presented by
Yan Yu
Advisor: James Glimm

2. Presentation Outline Introduction Error Analysis for Planar Geometry
Statistical Numerical Riemann Problem
Composition Law for Errors
Numerical Results
Error Analysis for Spherical Geometry
Error Decomposition
Conclusions
Future Objective

3. Part Ι
Introduction

4. Introduction Our goal:
To formulate and validate a composition law to estimate errors in the solutions of complex problems in terms of the errors from the simpler ones. To understand the relative magnitude of the input uncertainty vs. the errors created within the numerical solution.
References:
1. Statistical riemann problems and a composition law for errors in numerical solutions of shock physics problems, J. Glimm, J. W. Grove, Y. Kang, T. Lee, X. Li, Y. Yu, K. Ye and M. Zhao, SISC(2003)
2. Errors in numerical solutions of spherically symmetric shock physics problems, J. Glimm, J. W. Grove, Y. Kang, T. Lee, X. Li, Y. Yu, K. Ye and M. Zhao, Contemporary Mathematics (2004)
3. Error Analysis of Composite Shock Interaction Problems, T. Lee, Y. Yu, M. Zhao, J. Glimm, X. Li and K. Ye, Conference Proceedings of PMC2004 (2004)

5. Graphic View
Introduction
From:
Study error models for individual shock interactions
To:
Estimate errors in the complex multi-interaction problem

6. Uncertainty Quantification
Introduction
Related approaches:
An early focus: round off errors.
Asymptotic analysis of numerical solution errors.
The use of a posteriori error estimator.
Mapping of input uncertainty to output random variables.
Our approach:
We study the errors statistically, in a pre-asymptotic range.
We allow for errors generated within the solution process.

7. Main Steps
Introduction
Construct wave filters that decompose a complex flow into approximately independent elementary waves.
Determine solution error models for a compre-hensive set of elementary wave interactions.
Formulate a composition law that constructs the total solution error at any space-time point in terms of errors from repeated elementary interactions.

8. The Wave Filter Wave profiles are reconstructed:
For contact or shock waves:
For rarefactions and compressions:

9. Operation of A Wave Filter
contact or shock
rarefaction or compressions

10. “Statistical Numerical Riemann Problem”
SNRP
“Statistical Numerical Riemann Problem”
A statistical distribution of numerical incoming waves and starting states determines the SNRP.
The waves in the SNRP have a finite width and the solution algorithm in the SNRP has only finite accuracy.
The SNRP introduces errors in addition to propagating errors or uncertainty from input to output.

11. Error Analysis for Planar Geometry
Part ΙΙ
Error Analysis for Planar Geometry

12. List of Interactions (SNRPs)
Planar Geometry
Left frame shows the type and location of waves as determined by our
wave filter analysis for the complex multi-interaction problem.
Right frame is a schematic representation of the waves and the interactions. Ten Riemann problems are extracted from it.

13. Isolated Shock Waves
Left frame shows the expected narrow and time independent shock width (~ 2Δx).
Right frame shows an exponential approach of the numerical shock profile to its limiting values at x = ±∞. The straight line is the asymptote to the exponential convergence rate.

14. Isolated Contact Waves
Top: step down contact (flow from high density to low)
Contact width grows with a rate asymptotically
proportional to t1/3.
i.e. wc ~ cct1/3, and we find cc ~ 1.
Bottom: step up contact (the reverse)
We find wc ~ min{5, cct1/3}.
Difference: the spreading is associated with the up stream side of the contact, and the continued spreading depends on the up stream flow being subsonic. We used the MUSCL scheme here.

15. Multinomial Expansion for SNRP Output
First, we represent the wave properties as
Then, the multinomial expansion for the output is defined as (for wave strength):
Given a statistical ensemble of input and output values ωi and ωo, we use a least square algorithm to determine the best fitting model parameters α K,J.

16. Expansion Coefficients
Shock contact interaction
coef
variable
constant
ω1i
ω2i
model error L∞
STD
ω1o
-0.208
0.454
0.251
0.47%
0.001
ω2o
-0.042
0.000
0.912
0.03%
0.0001
ω3o
-0.286
1.004
0.346
0.30%
λ1o
2.184
-0.563
122%
0.240
λ2o
4.725
0.110
-1.466
0.67%
0.010
λ3o
2.197
0.068
0.106
5.35%
0.057
p1o
0.221
-0.014
0.023
27.1%
0.022
p2o
0.426
-0.092
1.78%
0.002
p3o
0.332
-0.004
-0.099
3.47%
0.005

17. Expansion Coefficients
Shock reflection
coef
variable
constant
ω1i
model error L∞
STD
ω1o
-0.002
0.716
0.014%
λ1o
2.291
-0.422
7.923%
0.062
p1o
0.060
-0.039
5065%
0.009
ω2o
0.057
0.0003
24.4%
0.005
λ2o
5.9
50%
0.7
wall errors

18. Expansion Coefficients
Contact reshock interaction
coef
variable
constant
ω1i
ω2i
model error L∞
STD
ω1o
0.282
-0.314
0.645
0.57%
0.0008
ω2o
0.013
0.819
0.118
0.20%
0.0003
ω3o
-0.128
0.143
0.458
0.41%
0.0004
λ1o
2.383
0.754
-1.307
5.47%
0.038
λ2o
0.909
0.011
0.216
1.00%
0.005
λ3o
3.619
0.151
-0.974
14.8%
0.138
p1o
0.242
0.043
0.042
10.4%
0.014
p2o
-0.036
0.045
0.066
75.5%
0.008
p3o
-0.447
0.078
15.7%
0.029

19. The Composition Law The composition law Domain of dependence
––– A formula for combining the wave interaction errors defined for single Riemann problems to yield the error for arbitrary points in the complex wave interaction problem.
Which pieces to add up?
The initial waves and errors located inside the domain of dependence.
Domain of dependence

20. Multi-path Integral
The composition law
The total error at the final time can be formulated as
G –– a planar graph with all the Riemann problems and traveling waves.
V –– the set of vertices of G, V=V(G) .
B –– a subset of V, the vertices of which are inside the domain of dependence.
Iv –– the interaction coefficient, taken from the table we obtained before.
dωB –– the multi-path propagator, a product of the individual propagator for
each single path.

21. Transmission of Position Errors
The composition law
For interaction with a wall:
For interaction of two incoming waves:

22. Errors in Fully Resolved Calculations
The composition law
Simulation
Prediction
mean wave strengths
ω1o
0.451
0.452
ω2o
0.704
0.703
ω3o
0.999
0.998
wave strength errors
Var(ω1o)
0.0008
Var(ω2o)
0.0019
0.0018
Var(ω3o)
0.0035
0.0036
Simulation
Prediction
wave width errors
λ1o
1.630
1.622
λ2o
3.636
3.635
λ3o
2.346
2.352
wave position errors
P1o
0.220
0.226
P2o
0.313
0.312
p3o
0.200
0.202
for interaction 1

23. Errors in Fully Resolved Calculations
The composition law
Simulation
Prediction
mean wave strengths
ω1o
0.713
0.714
wave strength errors
Var(ω1o )
0.0018
wave width errors
λ1o
1.868
1.859
wave position errors
P1o
-0.118
-0.092
for interaction 2

24. Errors in Fully Resolved Calculations
The composition law
Simulation
Prediction
mean wave strengths
ω1o
0.520
0.519
ω2o
0.674
ω3o
0.306
0.305
wave strength errors
Var(ω1o)
0.0009
0.0010
Var(ω2o)
0.0012
0.0013
Var(ω3o)
0.0004
Simulation
Prediction
wave width errors
λ1o
2.097
1.982
λ2o
5.027
4.918
λ3o
2.875
3.033
wave position errors
P1o
-0.097
-0.105
P2o
-0.003
0.013
p3o
-0.151
-0.134
for interaction 3

25. Error Analysis for Spherical Geometry
Part ΙΙΙ
Error Analysis for Spherical Geometry

26. Main Steps Spherical Geometry Composition Wave Filter Law Data
Analysis

27. List of Interactions (SNRPs)
Spherical Geometry
This plot shows the type and location of waves as determined by our wave filter analysis for the complex multi-interaction problem. It is also a schematic representation of the waves and the interactions. Five Riemann problems are extracted from it.

28. 1D Spherical Geometry
New issues: The solution is not constant between two waves. Waves do not have constant strength between interactions.
Causes: As the shock wave strengthens while moving inward, it generates a moving out rarefaction of the opposite family.
Challenge: Need a simple model for the growth of a wave as it moves radially inward or the decrease as it moves out.

29. Single Propagating Shock Waves
Inward Shock
The Guderley Solution
Left frame shows the Mach number vs. radius for a single inward propagating shock.
Right frame shows the same data plotted on a log-log scale.

30. Single Propagating Shock Waves
Outward Shock
Left frame shows the Mach number vs. radius for a single outward propagating shock wave starting at different radii r0.
Right frame shows the same data plotted on a log-log scale, the dashed lines represent the power law model.

31. Multinomial Expansion for SNRP Output
First, we represent the wave properties as
Then, the multinomial expansion for the output is defined as (for wave strength):
Given a statistical ensemble of input and output values ωi and ωo, we use a least square algorithm to determine the best fitting model parameters α K,J.

32. Expansion Coefficients
Shock contact interaction
coef
variable
constant
ω1i
ω2i
model error
STD
STD/ ωo
ω1o
19.521
2.501
0.860
0.954%
ω2o
0.374
0.200
0.0003
0.042
7.650%
ω3o
3.568
0.402
-0.045
0.009
0.463%
δ10
-2.039
-3.200
-0.01
0.157
0.174%
δ20
0.236
0.016
-0.002
0.021
3.825%
δ30
0.053
0.003
-0.001
0.0008
0.041%
λ1o
1.675
0.305
0.017
0.085
λ2o
7.093
0.482
-0.146
0.239
λ3o
2.829
0.302
-0.024
0.107
p1o
-0.247
0.242
0.005
p2o
0.643
0.065
-0.011
0.192
p3o
-0.042
0.062
0.004

33. Expansion Coefficients
Shock reflection
coef
variable
constant
ω1i
model error
STD
STD/ ωo
ω1o
5.606
1.137
0.468%
δ10
-3.27
0.031
0.112
0.045%
λ1o
1.221
0.018
0.099
p1o
0.474
0.001
0.012

34. Expansion Coefficients
Contact reshock interaction
coef
variable
constant
ω1i
ω2i
model error
STD
STD/ ωo
ω1o
0.097
-0.108
0.436
0.031
13.305%
ω2o
0.103
-0.192
1.168
0.007
1.116%
ω3o
0.988
0.195
-0.225
0.003
0.262%
δ10
-0.291
0.161
-0.468
0.017
7.296%
δ20
-0.067
0.142
-0.125
0.006
0.957%
δ30
-0.030
0.107
0.001
0.087%
λ1o
9.776
-6.372
5.091
0.484
λ2o
1.903
0.156
-0.677
0.534
λ3o
4.088
-1.401
1.549
0.168
p1o
4.782
-3.602
2.372
0.379
p2o
-0.453
0.409
-0.054
0.177
p3o
-0.199
-0.685
3.213
0.052

35. Composite Shock Interaction Problems
The composition law
The total error at the final time can be formulated as
G –– a planar graph with all the Riemann problems and traveling waves.
V –– the set of vertices of G, V=V(G) .
B –– a subset of V, the vertices of which are inside the domain of dependence.
Iv –– the interaction coefficient, taken from the table we obtained before.
dωB –– the multi-path propagator, a product of the individual propagator for
each single path.

36. Errors in Fully Resolved Calculations
The composition law
Simulation
Prediction
100 vs mesh
500 vs mesh
wave strength errors and propagated initial uncertainties
δ1o
0.04±2(0.03)
0.03±2(0.02)
0.01±2(0.02)
0.009±2(0.01)
δ2o
0.14±2(0.05)
0.12±2(0.02)
0.03±2(0.01)
0.03±2(0.008)
δ3o
-0.02±2(0.02)
-0.02±2(0.01)
-0.006±2(0.005)
-0.007±2(0.004)
mean wave width errors
λ1o
3.04
2.83
2.63
2.72
λ2o
5.36
6.11
5.56
6.08
λ3o
2.71
2.92
2.98
mean wave position errors
P1o
1.25
0.23
0.12
0.18
P2o
0.43
0.06
0.05
0.04
p3o
-0.73
-0.15
-0.08
-0.11

37. Decomposition of the Error
Left frame: the type and location of waves as determined by our wave filter analysis for the complex multi-interaction problem. Right frame: Schematic diagram shows different sources contribute to the error at the output of interaction 3

38. Main Results from Data Analysis
Pie charts show the relative importance of transmitted input uncertainty and created error (100 and 500 mesh units, respectively). The plots are based on the variance of error or uncertainty associated with each of the six wave integral paths contributing to the post reshock wave strength for the contact wave. The two paths associated with input uncertainty are shown in hatched gray and the others are solid gray scales.

39. Part ΙV
Conclusions

40. Conclusions
A simple linear model of solution error is sufficient for the study of a highly nonlinear problem.
A composition law for combining errors and predicting errors for composite interactions on the basis of an error model of the simple constituent interactions has been formulated and validated.
For spherically symmetric shock physics problems, the main difficulties encountered were the non-constancy of waves and errors between interactions. The errors grow by a power law in the radius for a inward moving shock. Similarly outward moving shocks and their errors weaken by a power law.

41. Conclusions
Continued
For planar case, although our formulism allows for statistical errors in the ensemble, in fact the dominant part of all errors studied were deterministic.
For spherical geometry, we observed that for a 500 cell grid, the dominant error comes from the initial uncertainty, while for the 100 grid over 75% of the error arises within the numerical simulations.
The wave filter performs well as a diagnostic tool, but its limitation lies in assuming well separated waves.

42. Part V
Future Objective

43. Future Objective Error analysis for 2D perturbed simulations
Start with the solution convergence of direct numerical simulations (DNS) through mesh refinement, regardless of different numerical algorithms.
Determine the solution sensitivity to mesh size, algorithm, mass diffusion and etc.
Conduct error analysis in 2D perturbed simulations (both single mode perturbed interface and multi-mode perturbed interface).
Code comparison. For instance, FronTier code (tracked and untracked), AMR code, and AMR code with curvilinear coordinate, etc.

44. Perturbed Interface Single-mode perturbation Multi-mode perturbation
Future work
Single-mode perturbation
Multi-mode perturbation

45. Preliminary Simulations
Future work
Without offset
With offset

46. Code Comparison
Future work
FronTier simulation
AMR simulation

47. Thank You