Many problems in applied mathematics involve the solution of a differential equation. Simple differential equations can be solved analytically: we can find a formula expressing the solution for any value of the independent variable. But most equations are nonlinear and this approach does not work; we must solve the equation by approximate numerical means. The big question is:
“Does the numerical solution resemble the true solution of the equation?”
The answer is: “Not necessarily”.
There are often specific criteria that must be satisfied to ensure that the answer `crunched out’ by the computer is a reasonable approximation to reality. Although the principles of numerical stability are quite general, they are best illustrated by simple examples. We will look at some of these below.
Smooth curve: True solution. Black dots: stable solution. Red dots: unstable solution (time step too large).
Continue reading ‘Does Numerical Integration Reflect the Truth?’
