I am working through a paper on implicit-explicit predictor-corrector scheme for non-linear parabolic PDE and having trouble understanding some concepts. I have this PDE:
$\frac{\partial u}{\partial t}$ = $\frac{\partial^2 u}{\partial x^2}$ + $f(t,x,u)$
After spatial discretization by central difference operator, I have this:
$\frac{du}{dt}$ = $Au$ + $g(t,u)$
with $A$ a tridiagonal matrix and $g(t,u)$ a non-linear function. Tha author goes for implicit treatment of $Au$ (stiff term) and explicit treatment of $g(t,u)$ (non-linear ) by way of an IMEX scheme and gets this s-step scheme:
$\sum_{j=0}^{s} a_{j}u_{n+j} = k\sum_{j=0}^{s} b_{j}Au_{n+j} + k\sum_{j=0}^{s-1} c_{j}g(t_{n+j},u_{n+j})$
The above scheme is the first thing I would like help as to how to get it? Or is it just something I have to take as being defined as such? I have read through other journals, particularly one by Urim Ascher but I only get the end product with no actual explanation / derivation.
Then the author goes on to use splitting to avoid using iterations. From splitting, he gets the following predictor-corrector scheme:
predictor: $(a_{s}I - b_{s}kA)\tilde u_{n+s} = \sum_{j=0}^{s-1}(-a_{j}u_{n+j}+kb_{j}Au_{n+j}+kc_{j}g(t_{n+j},u_{n+j}))$
corrector: $(a_{s}I - b_{s}kA)u_{n+s} = \sum_{j=0}^{s-1}(-a_{j}u_{n+j}+kb_{j}Au_{n+j}+kb_{j}g(t_{n+j},u_{n+j}))+kb_{s}g(t_{n+s},\tilde u_{n+s})$
Any insight as to how we get this PC pair would be of great help. Thanks. Here also I have seen previous works by J. Cash and others but I can't seem to get the above.
