1. Experimental RELAP5-3D Time Step Improvements Dr. George L Mesina RELAP5 International Users Seminar Nov 18-20, 2008 Idaho Falls, ID

2. Overview Background on old method Improvements Measures Results

3. Background RELAP5-3D major time step controls –Material Courant Limit –Truncation Error –Upper and lower time step limits –Time-targeting Must exactly reach plot, output, restart times –Some other controls via input not relevant Trip stop, dump, etc.

4. Background: Material Courant Limit Material Courant Limit (MCL) In a control volume: Δt i n is the time required for fluid to cross from the entrance to the exit of volume i on time step n. Larger time steps with semi-implicit method cause instabilities. Cannot exceed Δt i n in any volume. Flow region MCL is:

5. Background: Time Step Limits Upper and Lower Limits –The user establishes upper and lower time step limits via input. –The upper limit creates a minimum number of steps to complete a transient (if Δt i n is never cut). Mass Error –The deviation from perfect solution of the continuity equation. The time step is cut if mass error becomes excessive or Δt i n exceeds MCL.... Ideally.

6. Background: Time Step Control However, to allow faster code execution, selective violation of MCL was implemented –5 “bins,” S J = {Δt n J+5i | i = 1, 2,..., NVOL/5}, J = 1,2,3,4,5 –MCL J = min(S J ).Note: MCL 1 == MCL. –Δt n = MCL 2.Note: Opt 15 sets Δt n = MCL 1 = MCL. To hit time targets exactly, use halving and doubling. –Plots, minor & major edits, restarts are multiples of DTMAX. –RULE: Never bypass a target time. –If we had T=1.75, plot time=2.0, and Δt n = 0.5, the T+dt > 2.0. Would need to HALVE Δt n to hit plot time. In practice, this is controlled differently.

7. Background: Time Step Control To hit time targets exactly, only double on even time steps. Controlled by integer, NREPET. NREPET = Number of Δt n steps needed to reach next multiple of DTMAX. –Example: Δt n = 0.25*DTMAX, and the cumulative time is: T n = 5.25*DTMAX, NREPET n = 3. –Double Δt n implies halve NREPET n. In example, 3/2=1. Wrong! –Need Δt n+1 =0.25*DTMAX. Then T n+1 =5.5*DTMAX, NREPET n+1 =2. –Double now. Δt n+2 =0.5*DTMAX. NREPET n+2 =1.

8. Background: Halving & Doubling Halving and doubling algorithm based on DTMAX –Δt n = 2 -k DTMAX,0 < k < log 2 (DTMAX/DTMIN). –Halve if MCL or mass error condition violated. –Double if mass error “low” & Δt n < MCL/2. Combining DTMAX limit with selective MCL violation, the flow region MCL is given by:

9. Improvements Deficiencies & possible improvements –Should never violate MCL Many users run with exact MCL only. –Allow time steps other than 2 -k DTMAX. However, still hit time targets exactly –Allow code to run above DTMAX (user option). Must stay “safely” below MCL. Apply multiplier, m<1.0, such that Δt n < m*MCL

10. Integer Time-step All work was done in the context of the integer time- stepping revision of subroutine DTSTEP. –This work enables exact calculation of time in long-running transients –Important for working with coupled codes. To understand integer time-stepping, thing of your computer’s clock cycle. –If it is a 4 GHz machine, it performs 4,000,000,000 ticks per second. –Each tick is 1/4000000000 sec. –An integer, t, can count the ticks from 1 to 4000000000. –Floating point time T = t/4000000000.0

11. Time Targets A 4-step approach is used to hit time targets. –Check for target 3 steps in advance.

12. Approach MCL if above DTMAX DTMAX is often far below MCL If the user could allow violation of DTMAX, larger time steps could often be taken. –Propose making the option controlled on the time step card. –The user could turn it on and off to examine important parts of the transient as needed.

13. MCL vs Current Time-step (DTMAX) Typpwr

14. Typpwr Violating DTMAX

15. Selecting the MCL Multiplier, m Multiplying the MCL by safety value ensures the time step does not come “too close” to instability. A study was done by running numerous test models with a variety of values of m. –m = 0.5k, k = 1, 2,..., 20. –The input models were taken from the set of problems transmitted with the code. The study showed that generally,.85 <= m <=.95 was best. –In fact, m = 0.9 proved about the best choice.

16. Background b

17. b

18. Combining DTMAX violation and m Combined improvements –Sometimes the number of time steps is reduced significantly. –Sometimes there is no change. Note that use of excessively frequent time targets interferes with every improvement –Because code must reduce time step to hit target For Typical PWR, the reduction is most pronounced.

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20. Conclusions The existing algorithm for semi-implicit time- stepping was reviewed. Several improvements were suggested. Two improvements were combined and shown to allow significant reduction in code runtime These are the MCL multiplier and DTMAX violation It is proposed that these be made a user option.