Numerical ODEs: Euler, RK4, Multistep & Applications

TAKEAWAYS:

• ODEs in engineering: Real-world problems (e.g., motion, circuits, biology) are modeled by ODEs, which rarely have analytical solutions—numerical methods approximate solutions step-by-step.
• Euler’s method: The simplest single-step method, but inaccurate for stiff equations due to linear approximation (y_{n+1} = y_n + h·f(x_n, y_n)).
• Runge-Kutta (RK4): A 4th-order method balancing accuracy and computation (w_1 = h·f(x_n, y_n), w_2 = h·f(x_n + h/2, y_n + w_1/2), etc.), used in finance (option pricing) and physics simulations.
• Multistep methods (Adams-Bashforth): Use past values (y_{n+1} = y_n + h/24·(55f_n – 59f_{n-1} + 37f_{n-2} – 9f_{n-3})) for efficiency but require initial steps from single-step methods.
• Error analysis: Local truncation error (LTE) and global error grow with step size h; smaller h improves accuracy but increases computation time.
• Real-world tie-ins: Pathao’s ride-distance calculator (ODEs for trajectory), NTC’s power-grid stability (ODEs for load dynamics), and Daraz’s inventory forecasting (logistic growth models).

1. Why Numerical ODEs?

ODEs describe dynamic systems (e.g., dy/dx = f(x, y)), but only ~5% have closed-form solutions. Numerical methods approximate solutions by discretizing the domain.

Example: Population Growth

The logistic ODE models limited growth: $$\frac{dP}{dt}=rP\left(1-\frac{P}{K}\right),P(0)=P_0$$ where:

• $P(t)$ = population at time $t$,
• $r$ = growth rate,
• $K$ = carrying capacity.

A sigmoid (S-shaped) curve showing exponential growth tapering to $K$. (Image: Qef (talk), Public domain, via Wikimedia Commons)

2. Single-Step Methods

(a) Euler’s Method

Definition: Uses tangent-line approximation over small steps h. $$y_{n+1}=y_n+h\cdot f(x_n,y_n)$$ Worked Example: Solve $\frac{dy}{dx}=x+y^2$, $y(0)=1$ for $x=0.2$ with $h=0.1$.

Steps:

1. $y_1=1.0+0.1\cdot (0.0+1.0^2)=1.1000$
2. $y_2=1.1+0.1\cdot (0.1+1.1^2)=1.2100$

Error: Euler’s LTE is $O(h^2)$; global error $O(h)$.

(b) Runge-Kutta (RK4)

Definition: A weighted average of slopes at intermediate points for higher accuracy. $$\begin{array}{rl}{k}_1& =h\cdot f(x_n,y_n),\\ k_2& =h\cdot f(x_n+h/2,y_n+k_1/2),\\ k_3& =h\cdot f(x_n+h/2,y_n+k_2/2),\\ k_4& =h\cdot f(x_n+h,y_n+k_3),\\ y_{n+1}& =y_n+\frac{1}{6}(k_1+2k_2+2k_3+k_4).\end{array}$$ Worked Example: Same ODE, $y(0.2)$ using RK4.

Comparison Table:

3. Multistep Methods

Use past values to compute future steps (e.g., Adams-Bashforth). Adams-Bashforth 4-step: $$y_{n+1}=y_n+\frac{h}{24}\left(55f_n-59f_{n-1}+37f_{n-2}-9f_{n-3}\right)$$ Worked Example: Solve $\frac{dy}{dx}=-2xy$, $y(0)=1$ for $x=0.4$ with $h=0.1$.

flowchart TD
    A["Start: Obtain first 3 values with RK4"] --> B["Store f₀,f₁,f₂"]
    B --> C["Apply Adams‑Bashforth formula"]
    C --> D["Compute yₙ₊₁"]
    D --> E{"More steps?"}
    E -- Yes --> B
    E -- No --> F["Finish"]

Trace:

1. Use RK4 to compute $y_1,y_2,y_3$ (initial steps).
2. For $y_4$: $$f_3=-2\cdot 0.3\cdot 0.8825=-0.5295$$ $$y_4=0.8825+\frac{0.1}{24}\left(55(-0.5295)-59(-0.4500)+37(-0.3750)-9(-0.3000)\right)=0.7869$$

Advantages:

• Fewer function evaluations per step.
• Disadvantages: Requires starting values; unstable for stiff equations.

4. Error Analysis

Local Truncation Error (LTE): Error introduced in one step. Global Error: Accumulated over all steps. Stability: Methods like Euler may diverge for stiff ODEs (e.g., $\frac{dy}{dx}=\lambda y$, $\lambda \ll 0$).

In the Real World

1. Pathao’s Ride Distance Calculation

   • ODE Used: $\frac{d\vec{r}}{dt}=\vec{v}(t)$, where $\vec{r}(t)$ is position and $\vec{v}(t)$ is velocity (from GPS data).
   • Method: RK4 approximates the rider’s trajectory to compute distance and fare.
   • Example: For a ride from Thapathali to Lakshmi Narayan, Pathao’s backend solves this ODE with $h=0.01$ seconds to estimate the 3.2 km distance.

2. NTC’s Power Grid Stability

   • ODE Used: Swing equation for generator angles: $$\frac{d^2\delta }{d{t}^2}=\frac{\pi f_0}{H}\left(P_m-P_e\right),$$ where $\delta$ is rotor angle, $P_m$ is mechanical power, and P_e \.electrical power. - **Method**: Adams-Bashforth predicts load fluctuations to prevent blackouts. 3. **Daraz’s Inventory Forecasting** - **ODE Used**: Logistic growth for product demand: \[ \frac{dD}{dt} = rD \left(1 - \frac{D}{D_{\text{max}}}\right). \] - **Method**: Euler’s method forecasts stock levels (e.g., for iPhones) to trigger reorders. --- ### **5. Worked Example: Projectile Motion** **Problem**: Find the height of a ball thrown upward at \( v_0 = 20 \, \text{m/s} at $t=1\text{s}$, ignoring air resistance. ODE: $\frac{dv}{dt}=-g$, $v(0)=20$, $\frac{dy}{dt}=v$, $y(0)=0$.

Solution with RK4:

1. Define $y_1=v$, $y_2=y$: $$\frac{d{y}_1}{dt}=-9.8,\frac{d{y}_2}{dt}=y_1.$$
2. Use RK4 with $h=0.1$:
   • At $t=1.0$, $y_2\approx 18.62\text{m}$.

6. Choosing a Method

Exam Tip

1. Always show intermediate steps in RK4/Adams-Bashforth (examiners check $k_1,k_2,\dots$).
2. For stiff ODEs, mention stability conditions (e.g., "Euler fails for $\lambda h>2$").
3. Real-world questions: Relate to trajectory (Pathao), heat transfer (NTC grids), or growth models (Daraz inventory).
4. Error bounds: If asked for accuracy, state LTE/global error orders (e.g., "RK4 has LTE $O(h^5)$").
5. Graphs: Sketch solution curves (e.g., logistic growth) and error plots to visualize convergence.

Key Formula Cheat Sheet:

In the real world

• Pathao uses RK4 to integrate GPS‑derived velocity vectors, giving a smooth trajectory and accurate fare calculation.
• NTC (National Transmission & Dispatch Company) predicts generator rotor‑angle dynamics with a 4‑step Adams‑Bashforth predictor, enabling real‑time stability monitoring.
• Daraz forecasts product demand using Euler’s method on a logistic growth ODE, automatically triggering inventory re‑orders when the projected stock falls below a threshold.