Page 450 - Practical Design Ships and Floating Structures

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The so called flux vectors E and Fare evaluated in the n+l time level. Newton's linearization
procedure provides the flux vectors in the time level n:

Substituting (15) into equation (14), grouping AQ terms on the left-hand side of (14), and
approximating partial derivatives by central differences, it is possible to write down the following
equation:
(I + At(6.A" -6,MF)+ At(6,,B" - 8,M;)bQ" = -At(6$" + 6,,F") (16)
where
E" =E: -E" "9
F" = F;" - Fvn

6<( )= * , 6,( )= ( )Z,J+I -( It.J-1
( )S+l,J -( ),-I.,
2
Equation (1 6) is a penta-diagonal system of algebraic equations that solution can be improved by
convenient approximated factorization - Beam and Warming (1978). The left-hand side of (24) below,
after proper factorization of (16), presents only rand derivatives in separate terms. Therefore, the
numerical scheme of solution can be split into two steps. The first equation is solved for all interior
grid points; then, the second equation is solved for all interior grid points using the previous values of
f- In each step, the resulting block tridiagonal system of algebraic equations is solved by application of
Thomas algorithm (Anderson, Tannehill and Pletcher, 1984).

{Z + &(S, A - SsM6 ))" f " = (RHS)" (17)

+ A&B - 6,pn AQ~ fn
=
where (RH.)" =-~t(6~~ +6,~ (18)
Central difference schemes require artificial dissipation to improve stability - Pulliam (1980).
Artificial dissipation suppresses high frequency oscillations and controls the odd-even uncoupling
inherent to central difference schemes. The Von-Neumann linear stability analysis applied to the
Beam-Warming central difference scheme shows that some artificial dissipation is required to improve
stability.
(I + At6.A" - At6.M; + Df'P = (RHS)' + D'"
(1 9)
(I + &6,,B" - At6,,M; + D:)bQ'' = f"
where
0:) = -&,AU'V,A.,J; DF = -&iAtJ-'V,A,,& 0(4) -&&l?J-'[(V,A,p +(V,,A,)ZbP
=
3.2 Grid Generclrion

An algebraic grid generator using the multi-suface method - Fletcher (1988) - is used to generate the
grid points over and around the ship hull (body fitted grid). The grid is locally orthogonal to the body