Common partial differential equations of computational hydraulics 33
or in terms of the unknown value f i,j as
f
f
f
f
f
i j
i j
i j
i j
i j
,
,
,
,
,
=
+
+
+
(
)
+
−
+
−
1
4
1
1
1
1
(3.16)
This simple algebraic relation suggests that the numerical solution scheme
is implicit, requiring the synthesis and numerical solution of a system of
algebraic equations, referring to all the (i,j) locations inside the solution
domain. Equation 3.16 is not utilized at the boundary points where either
the values f(x,y) or the normal derivatives
df
dn
are known. One special feature of the resulting system of algebraic equations is the sparseness of the
constant coefficients matrix. The matrix is almost diagonal and can be easily solved by a successive iterations algorithm such as the easily programmable Gauss-Siedel iterative method.
3.3.1.1 Gauss-Siedel iterative method for solution
of linear algebraic system of equations
If the upper index (k) annotates the order of the iteration, then the computation of f i j
k
, values is accomplished starting from iteration (k = 1) and
continuing with second, third, and so on iterations, until the following convergence criterion is satisfied:
max ,
,
f
f
i j
k
i j
k
+
−
<
1
ε for all i,j
(3.17)
where ε is a very small predefined number. Using the notation in Equation
3.17, the Gauss-Seidel method can be described by the relation
f
f
f
f
f
i j
k
i j
k
i j
k
i j
k
i j
k
,
,
,
,
,
+
+
+
−
+
+
−
=
+
+
+
(
)
1
1
1
1
1
1
1
1
4
(3.18)
From Equation 3.18 it is evident that for the computation of f i j
k
,
+1 the solution domain is swept from the smaller towards the bigger i,j values using
the most recently corrected values of the function f(x,y). This is an easily
programmable procedure characterized by convergence to the correct value
of f, not counting numerical errors, but with an unknown number of iterations. The convergence is secured when the diagonal coefficient is bigger or
equal to the sum of the rest of the coefficients along a line of the coefficients
matrix. In the case under consideration (Equation 3.15) the coefficient 4 is
equal to the summation 1 + 1 + 1 + 1.
or in terms of the unknown value f i,j as
f
f
f
f
f
i j
i j
i j
i j
i j
,
,
,
,
,
=
+
+
+
(
)
+
−
+
−
1
4
1
1
1
1
(3.16)
This simple algebraic relation suggests that the numerical solution scheme
is implicit, requiring the synthesis and numerical solution of a system of
algebraic equations, referring to all the (i,j) locations inside the solution
domain. Equation 3.16 is not utilized at the boundary points where either
the values f(x,y) or the normal derivatives
df
dn
are known. One special feature of the resulting system of algebraic equations is the sparseness of the
constant coefficients matrix. The matrix is almost diagonal and can be easily solved by a successive iterations algorithm such as the easily programmable Gauss-Siedel iterative method.
3.3.1.1 Gauss-Siedel iterative method for solution
of linear algebraic system of equations
If the upper index (k) annotates the order of the iteration, then the computation of f i j
k
, values is accomplished starting from iteration (k = 1) and
continuing with second, third, and so on iterations, until the following convergence criterion is satisfied:
max ,
,
f
f
i j
k
i j
k
+
−
<
1
ε for all i,j
(3.17)
where ε is a very small predefined number. Using the notation in Equation
3.17, the Gauss-Seidel method can be described by the relation
f
f
f
f
f
i j
k
i j
k
i j
k
i j
k
i j
k
,
,
,
,
,
+
+
+
−
+
+
−
=
+
+
+
(
)
1
1
1
1
1
1
1
1
4
(3.18)
From Equation 3.18 it is evident that for the computation of f i j
k
,
+1 the solution domain is swept from the smaller towards the bigger i,j values using
the most recently corrected values of the function f(x,y). This is an easily
programmable procedure characterized by convergence to the correct value
of f, not counting numerical errors, but with an unknown number of iterations. The convergence is secured when the diagonal coefficient is bigger or
equal to the sum of the rest of the coefficients along a line of the coefficients
matrix. In the case under consideration (Equation 3.15) the coefficient 4 is
equal to the summation 1 + 1 + 1 + 1.