Given a mixed PDE such as \(u_{t}=Au+Bu\) where \(A,B\) are constant matrices.
Let standard stepping be
\[\begin {array} [c]{c}u^{\ast }=N_{A}\left ( u^{n},k\right ) \\ u^{n+1}=N_{B}\left ( u^{\ast },k\right ) \end {array} \]
Where \(N_{A}\) and \(N_{B}\) are numerical schemes to solve the problem \(u_{t}=Au\) and \(u_{t}=Bu\) respectively. \(k\) in the above is the time step.
Let Strang splitting be
\[\begin {array} [c]{c}u^{\ast }=N_{A}\left ( u^{n},k/2\right ) \\ u^{\ast \ast }=N_{B}\left ( u^{\ast },k\right ) \\ u^{n+1}=N_{A}\left ( u^{\ast \ast },k/2\right ) \end {array} \]
Now, assuming that \(N_{A}\) and \(N_{B}\) are each second order accurate in time. Which of the above two schemes should one select?
Algorithm
---- standard stepping IF A,B commute THEN standard stepping is second order in time ELSE standard stepping is first order in time END IF ---- Strang IF A,B commute THEN strang gives second order accuracy in time ELSE strang also gives second order accuracy in time END IF
Hence, from the above, the conclusion is that
IF A,B commute THEN select standard stepping (simpler) ELSE select Strang (more accurate) END IF
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