A new strategy for adapting time-step in the Local Time Stepping … · 2017-05-10 · Quang Huy Tran (IFP) Quang Long Nguyen . MAMCDP 2009 Workshop - Université Pierre et Marie

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P Extended reserves | Clean refining | Fuel-efficient vehicles | Diversified fuels | Controlled CO2

MAMCDP 2009 Workshop - Université Pierre et Marie Curie

A new strategy for adapting time-step in the Local Time Stepping Method applied to

hyperbolic PDEs

Équipe de Recherche Technologique: Frédéric Coquel, Marie Postel (LJLL) Quang Huy Tran (IFP)

Quang Long Nguyen

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Outline

Context Mathematical model and numerical scheme Multiresolution (MR) Local Time Stepping (LTS) Conclusion

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Outline



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Context

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Context

Transportation of hydrocarbons: oil and gas numerical simulation of two-phase flows in pipelines over a long

time-range Characteristics

Closure laws: costly to evaluate Time varying boundary conditions: inflow and outflow Pipeline's geometry: horizontal, slugging, severe slugging (gravity) fronts separating the oil and gas propagate along the pipeline

Goals Model the problem Improve the simulations Speed up the computation thanks to the Multiresolution and Local Time Stepping methods

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Outline



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Mathematical model

Drift-Flux Model

Closure laws Thermodynamics and Hydrodynamics :

nonlinear and costly

with

with

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Numerical scheme

Finite volume method with Riemann solver Explicit Lagrange-Projection scheme under the form

Under the CFL condition

avec

The positivity of the scheme is preserved


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Outline



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Multiresolution (MR)

Idea The mesh size depends on the local

smoothness of the solution adaptive grid The adaptive grid adapts dynamically

as the solution evolves in time grid prediction by detecting the

displacement and the formation of the singularities

Advantages Well adapted to our problem Reduce the number of calls to the

expensive closure laws gain in CPU time

Uniform

Adaptive

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Adaptive Multiresolution (MR)

Multiscale representation Mean value interpolation of the solution A. Cohen, S.M. Kaber, S. Müller and M. Postel, Math. Comp., 72

(2003), pp. 183–225

1. Encoding

2. Decoding h

h/2 level k-2

level k-1 level k

level k+1

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Multiresolution (MR) Thresholding algorithm

Details

Level dependent threshold parameter

Prediction (Harten's heuristic) Mark the neighborhoods of if

Mark the higher levels if

The adaptive grid is updated at each time step graded to ensure optimal complexity of the encoding/decoding

algorithm

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Numerical results (MR) Problem of shock tube flow


€

ρ

Yv

R

=

4000.4−10

€

ρ

Yv

L

=

5400.2−10

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Uniform solution versus MR solution Adaptive tree


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Convergence of the Multiresolution method

€

ε


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Outline



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Local Time Stepping (LTS)

Remark on MR Same time-step for all cells

Principle of the LTS Time-step adapted to mesh size

Advantage Further reduction of calls to closure

laws increase gain in CPU time

Adaptive Mesh Refinement

Local time stepping

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Local Time Stepping (LTS) Computation of the time-step

Definitions Macro time-step: corresponds to the biggest cells Micro time-step: corresponds to the smallest cells

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The micro time-steps decrease during each macro time-step They are updated along with the solution while ensuring the

stability condition

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The micro time-steps decrease during each macro time-step They are updated along with the solution while ensuring the

stability condition The first micro time-step

The others

where is the updated solution at time

The macro time-step

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Local Time Stepping (LTS) Synchronization in transition zones

?

? ?


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?

? ?

?


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S. Müller and Y. Stiriba, J. Sci. Comput., 30 (2007), pp. 493–531

?

? ?

?


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S. Müller and Y. Stiriba, J. Sci. Comput., 30 (2007), pp. 493–531

?

? ?

?


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LP conservative scheme Outside of transition zones

In transition zones


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Example of a mesh with 3 levels (K = 3) The solution is updated at each micro time step

0 1 2

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0 0 1 2

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0 1 2

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0 1 2

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Example of a mesh with 3 levels (K = 3) Synchronisation of the at the macro time step

0 1 2

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Numerical results (LTS) Problem of shock tube flow


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Numerical results - Convergence


€

ε

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Numerical results - Pipeline

Pipeline with slope of 30 degrees


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Sensors along the pipeline

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Time-steps’ evolution

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Performance

Comparison between two types of hydrodynamics laws


ratio UNI/MR ratio MR/LTS

N(K) Simple law Realistic law Simple law Realistic law

rCPU rNbLaw rCPU rNbLaw rCPU rNbLaw rCPU rNbLaw

256(5) 2,98 2,90 3,17 3,07 6,61 11,67 10,41 10,82

512(6) 4,19 4,64 5,02 4,85 7,77 18,79 16,13 17,06

1024(7) 5,47 6,99 7,48 7,13 8,39 29,20 24,58 25,88

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Outline



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Conclusion

Speed up the simulations thanks to the Multiresolution method Analysis of the local regularity of the solution Dyadic grid hierarchy Threshold parameter + prediction

Increase the performance by using the Local Time Stepping method coupled with MR Time-step adapted to the mesh size Faster, less calls to closure laws A new strategy to compute the time-step

The more expensive the closure laws are, the better the CPU gain is

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References [1] N. Adrianov, F. Coquel, M. Postel, Q-H. Tran A relaxation multi-resolution scheme for accelerating realistic two-phase

flows calculations in pipelines [2] A. Cohen, M.S. Kaber, S. Müller, M. Postel Fully adaptive multiresolution finite volume schemes for conservation laws [3] F. Coquel, Q-L. Nguyen, M. Postel, Q-H. Tran Local time stepping applied to implicit-explicit methods for hyperbolic systems [4] A. Harten Multiresolution algorithms for the numerical solutions of hyperbolic

conservation laws [5] S. Müller, Y. Stiriba Fully adaptive multiscale schemes for conservation laws employing

locally varying time stepping


http://www.ann.jussieu.fr/ERTint/

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Thank you for your attention


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Numerical method

Relaxation to treat the nonlinearities Replacing and by two new variables and Relaxation system

Hyperbolic system with linearly degenerate eigenfields

Subcharacteristic conditions

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Numerical method

Lagrange-Projection decomposition Based on ALE (Arbitrary Lagrange-Euler) formalism

3-step resolution


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Numerical method


3-step resolution


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Numerical method


3-step resolution


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Numerical method


3-step resolution


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Numerical method


3-step resolution


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Numerical method


3-step resolution


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Numerical method


3-step resolution


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Evolution of the adaptive tree


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Local Time Stepping (LTS) 1st strategy

Constant micro time-step during the whole simulation by using the smallest time-step issued from a uniform simulation

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Local Time Stepping (LTS) 2nd strategy

Constant micro time-step during each macro time-step by satisfying the very first stability conditions at time

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The first strategy stable but expensive as the time-step is very small needs to do simulation with a uniform grid first

The second strategy not sure to be stable as the solution evolves in time, i.e

after every micro time-step

A new strategy ??? ensure the stability


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LTS-fix


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Documents

A new strategy for adapting time-step in the Local Time Stepping … · 2017-05-10 · Quang Huy Tran (IFP) Quang Long Nguyen . MAMCDP 2009 Workshop - Université Pierre et Marie