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3 MacCormack Method
The MacCormack method is an explicit scheme that utilizes two step predictor–
corrector procedures for solving the governing equations at interior nodes. In
predictor step, backward difference is utilized for the spatial partial derivative
whereas in corrector step, forward difference is utilized for the spatial partial
derivative [4, 19].
3.1 General Formulation
In the predictor step, the partials are defined as under:
∂U
∂t
=
U
∗
i −U i
t
∂ F
∂ x
=
F i −F i−1
x
(4)
Substituting these values in the St. Venant equation, we obtain:
U
∗
i = U i −
t
x
(F i − F i−1 ) − t S i
(5)
The computed value U
∗
i gives A* and V *, which are used to compute F* and S*.
In the corrector step, the partials are defined as:
∂U
∂t
=
U
∗∗
i −U i
t
∂ F
∂ x
=
F
∗
i+1 −F
∗
i
x
(6)
Substituting these values in the St. Venant equations, the following is obtained:
U
∗∗
i = U i −
t
x
F
∗
i+1 − F
∗
i
− t S
∗
i
(7)
The value of the dependent variable at the next time unknown step j + 1 is finally
obtained as:
U
+
i =
1
2
U
∗
i + U
∗∗
i
(8)