Eikonal Blog

2010.01.26

Exponential splittings

Filed under: mathematics — Tags: — sandokan65 @ 12:06

First order splitting:

  • e^{t\sum_{i=1}^{n} A_i } = \prod_{i=1}^n e^{t A_i} + {\cal O}(t^2).

Second order splittings:

  • The Strang’s Splitting:
    e^{t\sum_{i=1}^{n} A_i } = e^{\frac{t}2 A_1} e^{\frac{t}2 A_2} \cdots e^{\frac{t}2 A_{n-1}} e^{t A_n} e^{\frac{t}2 A_{n-1}} \cdots e^{\frac{t}2 A_2} e^{\frac{t}2 A_1} + {\cal O}(t^3).
  • The parallel Splitting:
    e^{t\sum_{i=1}^{n} A_i } = \frac12(e^{t A_1}\cdots e^{t A_n} + e^{t A_n}\cdots e^{t A_1})  + {\cal O}(t^3).

Source: T1268 = Qin Sheng “Global Error Estimates for Exponential Splitting”; IMA Journal of Numerical Analysis (1993) 14, 27-56.

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