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# As long as convergence remains good the same jacobian

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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 6 A=6 6 4 3 a 7 a21 a (3.3.11) 7 ... ... . . . 7 : 5 as1 as2 a Thus, with the approximation (3.3.10), the system Jacobian in (3.3.10c) is 2 3 I ; haJ 0 6 ;ha21 J I ; haJ 07 6 (I ; hA J) = 6 .. 7 ... ... 7 : ... 4. 5 ;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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