However the derivation of such formula pairs is 46 not

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Unformatted text preview: unge-Kutta formulas for non-sti problems. His fourth- and fth-order formula pair is 0 1 4 1 4 3 8 3 32 9 32 12 13 1932 2197 7200 ; 2197 7296 2197 1 439 216 -8 3680 513 845 ; 4104 1 2 8 ; 27 2 ; 3544 2565 1859 4104 25 216 0 1408 2565 2197 4104 16 135 0 6656 12825 28561 56430 ^ ; 11 40 ;1 0 5 9 ; 50 2 55 The ^ denotes the coe cients in the higher fth-order formula. Thus, after determining ki, i = 1 2 : : : 6, the solutions are calculated as 25 1408 4 yn = yn 1 + h 216 k1 + 2565 k3 + 2197 k4 ; 1 k5] 4104 5 ; and 16 6656 9 2 5 yn = yn 1 + h 135 k1 + 12825 k3 + 28561 k4 ; 50 k5 + 55 k6]: 56430 ; Hairer et al. 16], Section II.4 give several Fehlberg formulas. Their fourth- and fth-order pair is slightly di erent than the one presented here. Example 3.5.5. Dormand and Prince 12] develop another fourth- and fth-order pair that has been designed to minimize the error coe cient of the higher-order method so that it may be used with local extrapolation. Its tableau follows. 48 0 1 5 1 5 3 10 3 40 9 40 4 5 44 45 56 ; 15 32 9 8 9 19372 6561 ; 25360 2187 64448 6561 212 ; 729 1 901...
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