SME
Numerical Methods for Engineers · Problem type
Step size for a stable Euler method
Use this for “For dy/dt = −50y, is Euler’s method with h = 0.05 stable? What step size is?”
Chart
Euler, 6 steps of h = 0.05: y = 11.39 at t = 0.3; solid, the exact y = 3.059 × 10⁻⁷, far off: the steps run away from it
This is an example. Type your own values to start a new problem.
g
=
1
+
(
k
)
×
h
hₘₐₓ
=
2
÷
|
k
|
y₀
Starting value
example
n
Steps
example
yₙ
Euler’s value after n steps
worked out (not typed)
Formulas
g
=
1
+
k
×
h
−1.5 = 1 + (−50) × 0.05
hₘₐₓ
=
2
÷
|
k
|
0.04 = 2 ÷ |−50|
yₙ
=
y₀
×
g
n
11.3906
=
1
×
(−1.5)
6
Assumptions
•
dy/dt = ky with k < 0 decays to 0, but each Euler step multiplies y by g = 1 + kh.
•
Stable when |g| ≤ 1, so h ≤ 2 ÷ |k|; past it the steps flip sign and grow.
•
A large |k| (a stiff problem) forces a tiny h; implicit methods avoid that limit.
Step-by-step
WE KNOW
k = −50, h = 0.05, y₀ = 1, n = 6
FIND
growth factor per step (g), largest stable step (hₘₐₓ), Euler’s value after n steps (yₙ)
Step 1 · Find growth factor per step (g)
Use g = 1 + k × h
An
Euler
step
adds
h
×
ky
to
y,
which
multiplies
y
by
1
+
kh.
g
=
1
+
(−50)
×
0.05
g
=
1
+
(−2.5)
g
=
−1.5
Step 2 · Find largest stable step (hₘₐₓ)
Use hₘₐₓ = 2 ÷ |k|
g
=
−1
at
h
=
2
÷
|
k
|;
any
larger
h
makes
g
<
−1.
hₘₐₓ = 2 ÷ |−50|
hₘₐₓ
=
2
50
hₘₐₓ
=
0.04
→
h
=
0.05
is
past
hₘₐₓ:
unstable;
any
h
≤
0.04
is
stable.
Step 3 · Find Euler’s value after n steps (yₙ)
Use yₙ = y₀ × gⁿ
n
steps
multiply
y₀
by
g
,
n
times.
yₙ
=
1
×
(−1.5)
6
yₙ = 11.3906
Check
Put the numbers back in. Both sides should match.
−1.5 = 1 + (−50) × 0.05 ✓
0.04 = 2 ÷ |−50| ✓
11.3906
=
1
×
(−1.5)
6
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Numerical ODE solvers (Euler, Runge–Kutta)
Main lesson
One Runge–Kutta (RK4) step
Use this for “Take one RK4 step of h = 0.5 on dy/dt = −0.5y from y = 10, and compare with the exact answer.”
One Heun step (improved Euler)
Use this for “Take one Heun step of h = 0.5 on dy/dt = −0.5y from y = 10, and compare with the exact answer.”