Let us now focus attention on the s x s Butcher tableau for the Implicit method. Hairer & Wanner quote Roger Alexander’s paper, where Alexander suggests the use of a lower-triangularized form of the table. By using the lower triangular form, the need for iterations vanishes and it becomes a single-step process.
In doing so, Alexander suggests that the term ai,j becomes 0
for all i < j. He refers to such a process as a Diagonally Implicit RK
method.
The Implicit RK formulation necessitates the solution of
ki’s via Newton-type iterations at each time step. A reformulation of the ki’s
can be written as follows (Hairer & Wanner):
Rearranging the terms to a form that has the independent
variable ki on the left hand side, the common term that arises in the iteration
is:
.png)
The partial derivative is equivalent to the Jacobian.
Alexander proceeds further to state that if all the ai,i
terms are equal, (i.e., the outer diagonal terms of the lower triangular
matrix), then the above matrix with the Jacobian needs to be evaluated only
once at each time step. Such a formulation is referred to as the Single
Diagonally Implicit RK method, or the SDIRK.
.png)
.png)
.png)
.png)
.png)
.png)
.png)
.png){kind=small}
.png)
.png)
.png)
.png)
.png){kind=small}
.png){kind=small}
.png)
.png)