Page 147 - Computational Fluid Dynamics for Engineers

Problems 133

4
1 11 d u
{u xx)i = -^{2ui - hui-i + Aui-2 ~ Uis) ~ 75 Ax2 ]Tl (P4.1.4)

4-2. Show that u xxx at i can be approximated by the following forward, back-
ward and central difference formulas.
Forward difference:
4
1 Ax 3 d u
{u xxx)i = -^-g(u i+3 - 3u i+2 + 3u i+i - Ui) J~Q~1: (P4.2.1)
or

(u xxx)i = 3(-3u i+4 + 14v,i+3 - 2Aui+ 2 + 18ifcj+i - 5^)

Backward difference:
4
1 Ax d u
{U xxx)i = -r-3 (l/i - 3 ^ _ i + 3Wj_ 2 - Wi-3) + ^ ~ ^ ~ 4 (P4.2.3)
or

(^xxx)i = 2 , 3 (5^i - 18izi_i + 24^_2 - 14ui_3 + 3^_ 4 )

- | ^ £ <P«-4)
Central difference:

5
1 1 9 i£
2
(uxxx)i = ^ - 3 (^z+2 - 2i/ i+ i + 2 ^ _ i - ixi_ 2) - -Ax —^ (P4.2.5)
or

(u xxx)i = 3 -^i+3 + %U i+2 - 13l/j+i + 13lXi_i - 8lZj_2 + Ui-s)
(

4-3. Show that u xxxx at z can be approximated by the following forward, back-
ward and central difference formulas.
Forward difference

l^l\ -( N
I dx 4 J ~ ^ Uxxxx ^
b
1 d u
r(u i+4
^ V ^ m - 4Ui+s + ' 6Ui+ 2 ~ 4 ^ + 1 + i Ui) - 2Ax—-^ ^ x 5 (P4.3.1)
WI-Y*
— z ^
™
-zy
z-hi