II. FINITE DIFFERENCE METHOD 1 Difference formulae
II. FINITE DIFFERENCE METHOD 1 Difference formulae
II. FINITE DIFFERENCE METHOD 1 Difference formulae
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Since A is invertible, the error of the solution can be evaluated 2 in norm:<br />
k E k=k A −1 τ k·k A −1 kk τ k (9)<br />
To ensure convergence two facts have to be verified:<br />
1) k τ k ! 0, for some norm: this property is the consistence of the<br />
numerical method<br />
2) k A −1 k· C, with C independent of h: this property is the stability<br />
of the numerical method, it is verified if the matrix A −1 is not divergent in<br />
norm as h ! 0.<br />
If we choose, for example, the norm k w k:= maxi jwij, we have<br />
k τ k= max jτij · Mh<br />
i<br />
2 /12 to 0 as h ! 0<br />
by (8), whence consistence is proved. For what concerns stability, one must<br />
prove that A is nonsingular uniformly with respect to h, for example by<br />
showing that the lowest eigenvalue is larger than a positive constant independent<br />
of h. That can be proved, the stability property holds, and the<br />
convergence of the method follows. 4<br />
3 Further applications of the difference method<br />
The creation of the world is, so to speak, a ”community project” of the Trinitarian<br />
God. The Father is the Creator, the Almighty, The Son is the meaning and heart of<br />
2<br />
In these evaluations, the norm of a vector can be the usual Euclidean norm (k w k=<br />
w2 1 + . . . + w2 n ), or the simpler (and topologically equivalent) maximum norm: k w k:=<br />
maxi jwij. The norm of a matrix is induced by the choice of the vector norm by the<br />
definition<br />
k Bv k<br />
k B k:= sup<br />
v=0 k v k .<br />
If, for example, the maximum norm for vectors is chosen, the induced norm of a matrix<br />
turns out to be the maximum among the sum of rows (in absolute value):<br />
k B k= max<br />
i<br />
n<br />
jbijj.<br />
In any case, after choosing a vector norm and the induced matrix norm, it is guaranteed<br />
that<br />
k Bv k·k B k ¢ k v k, 8v.<br />
12<br />
j=1