**Solution of the system of equations**

A Newton-Raphson iteration procedure was used to solve the system of equa-

tions given above. If an equation is designated as

(

)

*F *= *F d*j , ηj , *u*j , υj , *d*j+1 , ηj+1 , *u*j+1 , υj+1

(130)

then for any two points `*j*' and `*j *+ 1' at any time, it can be expanded as a Taylor

Series

*F*

*F*

*F*

*F*

*m*+1

*F*=*mF *+

∆ηj+1 +

∆ηj +

∆*d*j +

∆*d*j+1 +

ηj+1

ηj

*d*j

*d*j+1

*m*

*m*

*m*

*m*

(131)

*F*

*F*

*F*

*F*

∆υj+1

∆υj +

∆*u*j +

∆*u*j+1 +

υj+1

υj

*u*j

*u*j+1

*m*

*m*

*m*

*m*

where `*m*' and `*m *+ 1' indicate the iteration level and `*j*' and `*j *+ 1' indicate the *x*-

location. The partial derivatives are evaluated on the basis of the values of the vari-

ables after the `*m*th' iteration. Therefore, each equation can be transformed into a

linear equation in terms of the eight unknowns: ∆*d*j, ∆ηj, ∆*u*j, ∆υj, ∆*d*j+1, ∆ηj+1, ∆*u*j+1,

and ∆υj+1. The goal of the method is to solve for the change between `*m*' and `*m *+ 1'

such that m + 1F ⇒ 0. It can also be seen that mF is a function of the final values of the

variables at time `*n*' and the values for the `*m*th' iteration at time `*n *+ 1,' which are

all known values. The transformed equations are then put into the form

*K *=*m*+1*F*-*mF*

(132)

= *a*∆*d*j + *b*∆ηj + *c*∆*u*j + *d*∆υj + *e*∆*d*j+1 + *f*∆ηj+1 + *g*∆*u*j+1 + *h*∆υj+1

where

*F*

*F*

*F*

*F*

*a*=

*b*=

*c*=

*d*=

*d*j ,

ηj ,

*u*j ,

υj ,

*F*

*F*

*F*

*F*

*e*=

*f*=

*g*=

*h*=

,

ηj+1 ,

*u*j+1 ,

υj+1 .

*d*j+1

Since K =m+1F-mF , and the goal is to achieve

*m*+1

*F*⇒0

*K *=-*mF *.

(133)

For example, the conservation of water mass equation becomes

(d

) - (d

) + *u * θ(d

) + (1 - θ)(d

) +

*n*+1

*n*+1

*n*+1

*n*+1

*n*

*n*

*n*

*n*

j+1 + *d*j

j+1 - *d*j

j+1 + *d*j

j+1 - *d*j

*F*=

2∆*t*

∆*x*

(

)

(

) = 0

θ *u*n+1 - *u*n+1 + (1 - θ) un - *u*n

(134)

j+1

j+1

*d*

j

j

∆*x*

and the derivatives are evaluated as

47