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Efficient implementation of multirate time integration schemes - - PowerPoint PPT Presentation

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Efficient implementation of multirate time integration schemes Martin Schlegel Leibniz Institute for Tropospheric Research, Leipzig and Martin Luther University Halle-Wittenberg April 27, 2010 M. Schlegel (IfT, MLU) Multirate schemes April

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Efficient implementation of multirate time integration schemes

Martin Schlegel

Leibniz Institute for Tropospheric Research, Leipzig and Martin Luther University Halle-Wittenberg

April 27, 2010

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 1 / 19

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Contents

1 Motivation 2 Time integration 3 Spatial discretization 4 Implementation Issues

Time integration Data Exchange Parallelization issues

5 Simulation Results

Academic example Real Example

6 Conclusions and Outlook

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 2 / 19

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Motivation – MUSCAT

Multi Scale Atmospheric Transport model System for air pollution modeling Allows for complex simulations of atmospheric chemistry Online coupling to weather model possible Employs local refinement techniques Implementation basis: Written in Fortran 90/95 Communication via MPI Balancing via Metis/Parmetis

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 3 / 19

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Motivation – MUSCAT (cont.)

Local refinement

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 4 / 19

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1 Motivation 2 Time integration 3 Spatial discretization 4 Implementation Issues

Time integration Data Exchange Parallelization issues

5 Simulation Results

Academic example Real Example

6 Conclusions and Outlook

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 5 / 19

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Splitting of the RHS

System to solve: ∂w

∂t = G(w) nonstiff

+F(w) w1 = w (tn) ri =

i−1

  • j=1

(aij − ai−1,j) G (wj) vi (ci−1∆t) = wi−1 ∂vi ∂τ = 1 ci − ci−1 ri + F (vi) , τ ∈ [ci−1∆t, ci∆t] , i = 2, ..., s + 1 wi = vi (ci∆t) w (tn + ∆t) = ws+1

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 6 / 19

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Splitting of the RHS

w1 = w (tn) ri =

i−1

  • j=1

(aij − ai−1,j) G (wj) vi (ci−1∆t) = wi−1 ∂vi ∂τ = 1 ci − ci−1 ri + F (vi) , τ ∈ [ci−1∆t, ci∆t] , i = 2, ..., s + 1 wi = vi (ci∆t) w (tn + ∆t) = ws+1

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 7 / 19

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Decomposition by fluxes

Splitting of the advection operator Idea:

◮ Domain is organized in blocks ◮ Each block computes the fluxes leaving its cells

Necessary for parallelization Easier local refinement Problem: Block boundaries needed for computation Solution: Halo cells

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 8 / 19

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Flux splitting of the advection operator

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 9 / 19

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Flux splitting of the advection operator

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 9 / 19

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Flux splitting of the advection operator

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 9 / 19

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Mathematical properties

Recursive Flux Splitting Multirate scheme (RFSMR) Generic scheme Mass preservation Up to 3rd order of convergence (w.r.t. time step) Internal consistency

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 10 / 19

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1 Motivation 2 Time integration 3 Spatial discretization 4 Implementation Issues

Time integration Data Exchange Parallelization issues

5 Simulation Results

Academic example Real Example

6 Conclusions and Outlook

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 11 / 19

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Time integration

Splitting approach suitable for recursive implementation Advantages:

◮ Only little memory overhead ◮ Arbitrary number of temporal refinement levels

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 12 / 19

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Algorithm

for all stages i = 2, ..., s + 1 do ri =

i−1

  • j=1

(aij − ai−1,j) G (wj) compute local adv. fluxes exchange fluxes add received fluxes if ci > ci−1 then vi (ci−1∆t) = wi−1 compute system on next faster level

∂vi ∂τ = ...

compute local diffusion-reaction else add local net fluxes end if exchange concentration end do

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 13 / 19

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1 Motivation 2 Time integration 3 Spatial discretization 4 Implementation Issues

Time integration Data Exchange Parallelization issues

5 Simulation Results

Academic example Real Example

6 Conclusions and Outlook

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 14 / 19

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Program flow

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 15 / 19

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1 Motivation 2 Time integration 3 Spatial discretization 4 Implementation Issues

Time integration Data Exchange Parallelization issues

5 Simulation Results

Academic example Real Example

6 Conclusions and Outlook

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 16 / 19

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SLIDE 19
  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 17 / 19

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1 Motivation 2 Time integration 3 Spatial discretization 4 Implementation Issues

Time integration Data Exchange Parallelization issues

5 Simulation Results

Academic example Real Example

6 Conclusions and Outlook

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 18 / 19

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Conclusion

  • M. Schlegel (IfT, MLU)

Multirate schemes April 27, 2010 19 / 19