Unformatted text preview: ue is known as a singly diagonally implicit
Runge-Kutta (SDIRK) method. Thus, the coe cient matrix of an SDIRK method has
26 the form 2
4 3 a
... ... . . . 7 :
Thus, with the approximation (3.3.10), the system Jacobian in (3.3.10c) is
I ; haJ
6 ;ha21 J I ; haJ
(I ; hA J) = 6 ..
... 7 :
;has1 J ;has2J
I ; haJ
The Newton system (3.3.10) is lower block triangular and can be solved by forward
substitution. Thus, the rst block of (3.3.10c) is solved for Y1 ) . Knowing Y1 the
second equation is solved for Y2 ), etc. The Jacobian J is the same for all stages thus,
the diagonal blocks need only be factored once by Gaussian elimination and forward and
backward substitution may be used for each solution.
The implicit midpoint rule (3.3.2) is a one-stage, second-order DIRK method. We'll
examine a two-stage DIRK method momentarily, but rst we note that the maximum
order of an s-stage DIRK method is s + 1 2].
Example 3.3.5. A two-stage DIRK formula has the ta...
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- Spring '14
- Numerical Analysis, yn, Tn, Numerical ordinary differential equations