The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 4. IMPROVING ON EULER’S METHOD 79
Figure 4.4: The region of absolute stability for Euler’s method is indicated by the
gray area.
To find the eigenvalue of equation 4.80 we linearize. Since the system is scalar, the
Jacobian reduces to a single partial derivative ∂f/∂y. For f(t, y) = −5ty 2 +5/t−1/t 2
we have the linearized equation
y ′ ≈ f y (1, 1)y (4.94)
)
=
(−10(t)(y)| t=1,y=1
y (4.95)
= −10y (4.96)
Hence λ = −10 and we need h ≤ −2/(−10) = 0.2 to remain in the region of absolute
stability. Of course, this eigenvalue will change as the solution progresses, and values
of t and y change, so the step size needs to be adjusted dynamically to remain within
the desired region.
4.3 Stiffness and the Backward Euler Method
An initial value problem is said to be stiff if the absolute stability requirement
leads to a much smaller value of h than would otherwise be needed to satisfy the
accuracy requirements. Thus stiffness depends on three factors: (1) accuracy; (2)
the length of the interval of integration; and (3) the region of absolute stability of
the problem. An example is given in figure 4.5. From equation 4.92 this problem,
which has λ = −100 requires h < 0.02 for small t. For the test equation the problem
is stiff on the interval [0, b] if
bRe(λ)