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Splitting Methods (Supp) (Lie Trotter splitting (Ch1) (3 steps for…
Splitting Methods (Supp)
Intro (Ch0)
operator exponentials
bounded \( \Rightarrow \) convergent series
\( e^A=\sum\limits^{\infty}_{n=0} \frac{1}{n!}A^n\)
unbounded \( \Rightarrow \) exp. is solution of diff. eq.
\( A \) elliptic \( \Rightarrow e^{-tA}\) bounded operator
ODE \( u'+Au+Bu=0\)
linear/non-linear operators
IC
splitting methods
combine existing solvers
sub-problems
Predictor corrector methods (Ch3)
methods which preserve stationary solutions
bounded operators \( \Rightarrow O(\tau^m)\)
Variation of constants (Ch2)
ODE to equivalent integral eq.
use numerical int. methods
Err ana = consistency err for the num. int. + stability of discrete evolution
unbounded \( A \) & bounded \( B \)
3 options for the next time step
left-sided rectangular rule = 1 step expl. Euler + solve simpler ODE
implicit midpoint rule = combi
right-sided rectangular rule = solve A-ode + 1 step of implicit Euler
Lie Trotter splitting (Ch1)
3 steps for \(u(t+1)\)
exact flow \( e^{-\tau(A+B)} \)
Strangs splitting w/ evolution \( e^{-\frac{\tau}{2}A }e^{-\tau B}e^{-\frac{\tau}{2}A }\)
& truncation err \( O(\tau^3)\)
\( A,B \) bounded \( \Rightarrow e^{-\tau(A+B)}-e^{-\tau A}e^{-\tau B}=O(\tau^2) \) (Le1)
discrete evolution \( u_{n+1}=e^{-\tau B}e^{-\tau A}u_n \)
\( A,B\) commute \( \Rightarrow \) splitting exact \( e^Ae^B=e^{A+B}\)
Examples (Ch4)
Alternating directions method (ADI) (Ch4.3)
convection diffusion eq (Ch4.1)
N-S-eq.s (Ch4.4.)
Schrödinger eq. (Ch4.2)