I have a 3D diffusion equation that I want to solve using a time splitting (2D+1D). Assume that $A$ is the 2D discrete diffusion operator and $B$ is the 1D discrete diffusion operator.
I want to use a STRANG splitting to ensure a global 2nd order of precision : \begin{equation} \frac{\partial f}{\partial t}=\frac{1}{2}Af\\ \frac{\partial f}{\partial t}=Bf\\ \frac{\partial f}{\partial t}=\frac{1}{2}Af\\ \end{equation} And for each time integration, I want to apply the BDF2 implicit formula to ensure stability and a local 2nd order precision: \begin{equation} \frac{\partial f}{\partial t}\approx \frac{f^{n+1}-\frac{4}{3}f^n+\frac{1}{3}f^{n-1}}{\frac{2}{3}\Delta t} \end{equation} My question is : for the time iteration $n\Delta t$, what is the $f^{n-1}$ of each stage of the splitting ? Is it the state of $f$ at the previous time iteration or the previous splitting stage ?
To be more literal, do I follow this algorithm ? \begin{equation} f0,f1\rightarrow(\frac{\Delta t}{2} A) f21\\ f1,f21 \rightarrow(\Delta t B) f22\\ f21,f22\rightarrow(\frac{\Delta t}{2} A) f2 \end{equation} and for the following iteration we initialize $f0=f22$ and $f1=f2$.
Or this one ? \begin{equation} f0,f1\rightarrow(\frac{\Delta t}{2}A) f21\\ f0,f21 \rightarrow(\Delta t B) f22\\ f0,f22\rightarrow(\frac{\Delta t}{2} A) f2\\ \end{equation} and for the following iteration we initialize $f0=f1$ and $f1=f2$.
Is there another way to ensure an unconditionnaly stable time integration splitting with a global and local 2nd order ?
