Systems of Ordinary Differential Equations - Elements of Scientific Computing - page 89

F nC1 F n
t

1
2 Œ.1 S n / C .1 S nC1 / ;

D

(3.55)

S nC1 S n
t

1
2 Œ.F n 1/ C .F nC1 1/ :

D

This system can be rewritten as

F nC1 C 2 D F n C t 2

S nC1 S n ;

(3.56)

2 D S n t C 2

F nC1 C S nC1 F n :

Note that for each n, this defines a 2 2 system of linear equations. Let us make this
clearer by defining the matrix
A D 1t 2
t=2

;

(3.57)

1

and the vector

b n D F n C t

:

t
2

S n

(3.58)

t
2

S n t C F n

Then system ( 3.56 ) can be written in the form

Ax nC1 D b n ; (3.59)

where x nC1 denotes a vector of two components. By solving the linear system
( 3.59 ), we find x nC1 and define

F nC1
S nC1

D x nC1 :

(3.60)

We note that since

det. A / D 1 C t 2 =4; (3.61)

we have det. A />0for all values of t and the scheme ( 3.56 ) is therefore always
well defined.
In general, a 2 2 matrix

B D ab
cd

(3.62)

is non-singular if ad ¤ cb, in which case the inverse is given by

d b
ca

:

1
ad bc

B 1 D (3.63)