Page 13 - Microsoft PowerPoint - Galkin-APS-DPP-2020-talkFinal

... and adaptive time stepping
𝜖􏰁􏰂􏰃 = max(𝑅 𝑥̅􏰁􏰂􏰃 − 𝑅 𝑥̅􏰁 )
Criterion for the convergence - difference between consequent residuals.
Typically level of ε ~10-16 (machine 􏰄
accuracy) can be reached for 5-6 iterations.
• Adaptive time stepping is crucially important for stiff systems
• Evolutionary Proportional Integral Derivative (PID) controller adjusts time steps smoothly to approach the prescribed tolerance, basing on relative changes, denoting un+1= 𝑥􏰀, un=x,..., then:
en = ||un+1−un||/||un+1||
􏰅 􏰉􏰊􏰋􏰌􏰍􏰉􏰎 􏰅􏰏 􏰉􏰐
Relative changes Adapted time step
Empirical PID parameters
P R I V I L E G E D
d𝑡􏰁􏰂􏰃= 􏰆􏰇􏰈 􏰆􏰇􏰈 d𝑡􏰁 􏰅􏰆 􏰅􏰆 􏰅􏰆 􏰅􏰆􏰇􏰏
kP = 0.075, kI = 0.175, kD = 0.01
Presenter: Sergei Galkin
A N D
C O N F I D E N T I A L 13