Numerical solutions and error

Chapter 10 - Follow the local rule in finite steps

Many useful differential equations have no solution expressible by a finite combination of familiar elementary functions. Numerical methods turn the rate rule into a sequence of approximations.

Euler's method uses the current slope for one short step:

$$\boxed{y_{n+1}=y_n+h f(t_n,y_n)},\qquad t_{n+1}=t_n+h.$$

It is the tangent-line approximation repeated.

Lab: shrink the step

For each equation, compare $h=1$, $0.5$, $0.25$, and $0.1$. Smaller steps usually reduce error, but they require more work. "Numerical" does not mean "unreliable"; it means error must be estimated and controlled.

Worked example: show every step

A table is part of the reasoning

The table prevents a common error: evaluating the new slope at a mixture of old and new coordinates.

Local and global error

Euler replaces a curved segment by a tangent step. If the exact solution has bounded second derivative, one step's local truncation error (the unscaled one-step defect from exact starting data) is $O(h^2)$. Some texts divide this defect by $h$ and call the result local truncation error instead. If $f$ is also Lipschitz in the state near the solution, accumulation across a fixed time interval gives global error $O(h)$. These are upper-order bounds; special solutions can have zero error.

Practical consequence: halving $h$ should roughly halve Euler's global error once the steps are small enough for this small-step pattern to appear.

Better slopes: midpoint and Runge-Kutta

Euler uses the slope at the start of a step. A midpoint method samples a slope near the middle. Classical fourth-order Runge-Kutta blends four slope samples. The central design idea is the same: spend more rate evaluations to approximate the curve across each step more accurately.

You do not need to memorize this fourth-order method, often called RK4, here. You should be able to ask:

• where does the method sample slopes?
• what order of global error does it claim?
• does halving the step produce the expected convergence?
• does the numerical curve respect known equilibria, signs, and bounds?

Numerical stability is not physical stability

For $y'=-10y$, the true solution decays. Euler gives

$$y_{n+1}=(1-10h)y_n.$$

If $h=0.3$, the multiplier is $-2$: the approximation alternates and grows even though the true system decays. The step is too large for Euler's stability region.

Fade the computation

Use Euler with $h=0.25$ for $y'=y(1-y)$, $y(0)=0.2$.

1. $y_1=0.2+0.25[0.2(0.8)]=\underline{\hspace{4em}}$.
2. Evaluate the next slope at $(t_1,y_1)=(0.25,\underline{\hspace{3em}})$.
3. Compute $y_2$ and confirm that the approximation stays between 0 and 1.

Practice

1. Practice - Euler table

   10.1 Use Euler with $h=0.2$ to approximate $y(0.6)$ for $y'=y-t$, $y(0)=1$. Show a slope table.

   Detailed solution

2. Practice - convergence

   10.2 For $y'=y$, $y(0)=1$, derive Euler's approximation $y_n=(1+h)^n$. Compare $h=1,1/2,1/4$ at $t=1$ with $e$.

   Detailed solution

3. Integrated - qualitative check

   10.3 Design a test that would catch a numerical solution of $P'=P(1-P/10)$, $P(0)=2$, crossing above 10.

   Detailed solution

4. Challenge - step stability

   10.4 For $y'=-ay$ with $a\gt 0$, find all $h\gt 0$ for which Euler approximations decay in magnitude. Which subset decays without alternating sign?

   Detailed solution

5. Practice - extension

   10.5 The explicit midpoint method uses $k_1=f(t_n,y_n)$, $k_2=f(t_n+h/2,y_n+hk_1/2)$, and $y_{n+1}=y_n+hk_2$. Apply two steps with $h=0.5$ to $y'=y-t,\ y(0)=0$. Compare $y(1)$ with two Euler steps and the exact value from $y=t+1-e^t$.

   Detailed solution

6. Challenge - transfer

   10.6 For $z'=rz(1-z)$, $r\gt 0$, Euler gives $z_{n+1}=z_n+hrz_n(1-z_n)$. Show that $0\lt z_n\lt 1$ implies $0\lt z_{n+1}\lt 1$ whenever $0\lt hr\le1$. Test $r=1,\ h=2,\ z_0=0.75$ to demonstrate overshoot, and explain why a bound-preserving step is not automatically accurate.

   Detailed solution

Retrieve the whole loop

Choose one equation from any earlier chapter and state:

1. its structural fingerprint;
2. a qualitative prediction;
3. an analytic method, if available;
4. a numerical fallback; and
5. two independent checks.

I compared at least four step sizes and checked the numerical results against qualitative behavior.

Next: Enter the mixed practice studio, where the method is never named for you.