Page 12 - Microsoft PowerPoint - Galkin-APS-DPP-2020-talkFinal

Semi-implicit time integration scheme
𝑡
𝑡̂ = 𝑡 + 𝑑𝑡
𝑥􏰀
𝑥̅ = 0.5(𝑥􏰀 + 𝑥)
𝑥􏰀 − 𝑥 = 𝐹􏰁 = 𝐹 𝑥 ̅ 𝑑𝑡
𝐹𝑥̅ =𝐹𝑥 +𝐽𝑥 𝑥̅−𝑥 +𝑅(𝑥̅)
𝑥􏰀􏰂􏰃􏰄 = 𝑥 + 𝐼 − 0.5𝑑𝑡𝐽 􏰅􏰄𝑑𝑡(𝐹 + 𝑅(𝑥̅􏰂))
Centered scheme, with Ơ(dt2) order of approximation (like Crank–Nicolson)
Expansion in a vicinity of x through Jacobian and non- linear residual R
Semi-implicit scheme uses iterations between t and 𝑡̂ levels, 𝑥􏰀􏰂􏰃􏰄, loop on n=0,1,2... (I - identity matrix)
current time level with x - known value dt - time step next time step level
unknown on next time level intermediate level
𝑅𝑥􏰀􏰂􏰃􏰄 =𝐹𝑥̅􏰂􏰃􏰄 −𝐹−𝐽(𝑥̅􏰂􏰃􏰄−𝑥)
Residual corrections starts with R=0 at n=0 until converge Presenter: Sergei Galkin P R I V I L E G E D A N D C O N F I D E N T I A L 12