Differential Equations: The First Steps - Elements of Scientific Computing - page 46

Convergence

We want, of course, the numerical scheme to be convergent in the sense that the
numerical solution converges toward the analytical solution as t approaches zero,
or, equivalently, as N goes to infinity. For the simple model considered here, we can
prove convergence in a direct manner. With a D 1 and T D 1; we have

1
N / N ;

r.1/ r N D .1 C

and since 14

1
N / N D e D r.1/;

lim
N ! 1

.1 C

it follows that r N converges to r.1/ as N goes to infinity.

2.2.3 Numerical Stability

Let us now consider the initial value problem

y 0 .t / D 100y.t/; (2.30)

y.0/ D 1;

which has the analytical solution

y.t/ D e 100t :

Let us try to solve this problem numerically from t D 0 to t D 1 by the method
introduced above. We let y n denote an approximation to y.t n / where, as usual,
t n D nt and t D 1=N: The numerical approximation is defined by the finite
difference scheme

y nC1 y n
t

D 100y n ;

so

y nC1 D .1 100t/y n : (2.31)

By reasoning as above, we get

y n D 1

n :

100
N

14 See your calculus textbook.