Towards GrAALE error estimation and grid adaptation

1
Towards GrAALE error estimation and grid
adaptation

• Giovanni Lapenta

2
Outline

• GOAL we want to have an operational way to
  detect the error of a given numerical
  discretization scheme
• How to define the error
• method
• example
• results
• Application to grid adaptation
• Current/Future directions

3
How to define the error

• Theoretical issue what is the most convenient
  definition
• Practical issue how to compute it in practice

4
Residual vs Actual Error

• Truncation Error Error Source
• Error propagates hyperbolic equation for the
  error
• Adaptation to error source IS SUPERIOR to
  adaptation to actual error

Zhang, Trepanier, Camarero, Comput. Meth. Appl.
Mech. Engrg. 185 (2000) 1-19
5
Error Source is best guide

• This test problem illustrates that an efficient
  grid adaptation can be achieved using the
  estimated error source instead of the error
  distribution itself. (Zhang, Trepanier, Camarero)

Uniform
Adapt to error
Adapt to error source
6
(No Transcript)
7
(No Transcript)
8
(No Transcript)
9
Consistency of Error Detectors

• Given an error detector and a discretization
  scheme.
• Apply Taylor series analysis to the error
  detector
• Apply MEA to the discretization scheme.
• Compare the leading term in the ME with the error
  detector
• An error detector is consistent if the two have
  the same order AND the same multiplier (save a
  constant factor).

10
Example Advection Equation

• We solve it with the Upwind method

11
Recipe for Error Detection
12
(No Transcript)
13
Gradient Recovery
M Ainsworth, JT Oden, A posteriori error
Estimation in Finite Element Analysis (Wiley, New
York, 2000)
14
The error detector is consistent
15
(No Transcript)
16
(No Transcript)
17
(No Transcript)
18
Study of Practical Errors

• Benchmarks of the Error Detectors
• 1D shock tubes
• 2D Colellas wedge problem
• Adaptation based on Error Detection
• Debye-Huckel equation
• Gas Dynamics

19
LAGRANGIAN GAS DYNAMICS

• Shock tube
• Sample Lagrangian run
• Error Detector

Lagrangian
20
(No Transcript)
21
(No Transcript)
22
LAGRANGIAN GAS DYNAMICS

• Error accurate in
• Contact Discontinuity
• Shock
• Rarefaction
• Operator recovery error successful

23
EULERIAN GAS DYNAMICS

• Shock tube
• Sample ALE run
• Error Detector

ALE
24
(No Transcript)
25
(No Transcript)
26
EULERIAN GAS DYNAMICS

• Colellas wedge problem
• M10 shock flowing towards a 30 degree wedge
• CLAWPACK (LeVeque)

27
Solution
Error detector
28
EULERIAN GAS DYNAMICS

• Operator recovery error successful
• Warp indicator used in some LANL AMR codes fail

29
How to adapt the gridbased on the error defined
above
30
Error Minimization

• Define a mapping
• To minimize the error as measured by the
  detector
• Submitted to JCP

31
Applications of Grid Adaptation

• Elliptic
• Debye-Huckel
• Hyperbolic
• Gas Dynamics

32
Test Debye-Huckel

• Grid Adaptation based on
• Error Detector
• Heuristic guidance
• Case considered

F
33
Heuristic vs Error Detector
Error detector
Heuristic
34
Why the heuristic grid fails
35
Convergence Rate
s.1
36
Convergence Rate
s.01
37
Gas Dynamics Nohs Benchmark
38
How to include time discretization also?

• By interpolation reconstruction also in time
• Collaboration with Alex Krganov

39
(No Transcript)
40
(No Transcript)
41
(No Transcript)
42
The error detector is consistent
And it includes the time discretization
error
43
Convergence Rate Lagrangian
Red Actual error Green Kurganov Yellow
Lapenta
CFL0.25
44
Convergence Rate Eulerian
Red Actual error Green Kurganov Yellow
Lapenta
CFL0.25
45
Conclusions

• Three main achievements
• Reliable practical way to detect the error
• The error is consistent with the MEA
• Time AND Space errors are included
• Two main benefits
• Practical improvement of ALE and AMR
• Application to VV

46
(No Transcript)
47
(No Transcript)
48
(No Transcript)
49
(No Transcript)
50
(No Transcript)