Initial Value Problems - Numerical Methods in Engineering with MATLAB - page 274

Stabilityis determinedbythree factors: the differentialequations, the method of
solution and the value of the increment h . Unfortunately, it is not easy to determine
stabilitybeforehand, unless the differentialequationislinear.

Stability of Euler's Method

As a simple illustration of stability,consider the problem

y =− λ y y (0) = β (7.11)

where λ is apositiveconstant. The exact solution of this problemis

e − λ x

= β

y ( x )

Let us now investigate whathappens whenwe attempttosolve Eq. (7.11) numer-
icallywith Euler'sformula

hy ( x ) y ( x + h ) = y ( x ) + (7.12)

Substituting y ( x ) =− λ y ( x ), we get

y ( x + h ) = (1 − λ h ) y ( x )

If | 1 − λ h | > 1, the methodis clearlyunstable since | y | increases in every integration

| − λ | ≤

step. Thus Euler's methodisstable onlyif 1 h 1,or

h ≤ 2 /λ (7.13)

The results can beextended to a system of n differentialequations of the form

y =− y (7.14)

where is a constant matrix with the positiveeigenvalues λ i , i = 1 , 2 ,..., n . It can be

shown that Euler's implicit method of integration formulaisstable onlyif

h < 2 /λ max (7.15)

where λ max is the largest eigenvalueof .

Stiffness

An initial value problemiscalled stiff ifsometerms in the solutionvector y ( x ) vary
much more rapidlywith x than others.Stiffness can beeasilypredicted for the differ-
entialequations y =− y with constant coefficient matrix . The solution of these

= i C i v i exp(

equations is y ( x ) and v i are

the corresponding eigenvectors. It isevidentthat the problemisstiff if there is a large
disparityinthe magnitudes of the positiveeigenvalues.

− λ i x ), where λ i are the eigenvalues of