Ordinary Differential Equation-2: 1st Order ODE (RK4 Method) - Computational Physics Formulation & Simulation

Program 1

1st Order ODE (RK4 Method)

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Mathematical Problem Formulation

An object in free fall under the influence of gravity, with constant acceleration: $a = 9.81 \, \text{m/s}^2$

$$ \frac{dv}{dt}=a ;\quad v(t=0)=0 $$

The following code implements the Runge-Kutta 4th order (RK4) method to calculate the velocity of the object over a time period.

Theoretical Background & Computational Mechanics

Runge-Kutta 4th Order (RK4) Method

The RK4 method is applied to a differential equation of the form:

$$ \frac{dy}{dt} = f(t, y) $$

with an initial condition $ y(t_0) = y_0 $ over a time interval $[t_0, t_{\text{end}}]$. The method proceeds as follows:

Steps of RK4

For each time step $ h $, the RK4 method calculates four slopes:

Compute $ k_1 $: This is the initial slope, calculated at the beginning of the interval.

$$ k_1 = f(t, y) $$
Compute $ k_2 $: This slope is calculated at the midpoint, using $ k_1 $ to approximate the halfway position.

$$ k_2 = f\left(t + \frac{h}{2}, y + \frac{h}{2} \cdot k_1\right) $$
Compute $ k_3 $: Similar to $ k_2 $, but uses the slope from $ k_2 $ to further refine the midpoint estimate.

$$ k_3 = f\left(t + \frac{h}{2}, y + \frac{h}{2} \cdot k_2\right) $$
Compute $ k_4 $: This final slope is calculated at the end of the interval.

$$ k_4 = f(t + h, y + h \cdot k_3) $$

Update Rule

The next value, $ y_{\text{next}} $, is computed as a weighted average of these slopes:

$$ y_{\text{next}} = y + \frac{h}{6} \cdot (k_1 + 2 \cdot k_2 + 2 \cdot k_3 + k_4) $$

Theoretical Principles & Computational Physics Guide

1st Order ODE (RK4 Method): Differential Equations & Dynamical Simulation

Dr. Alorika Chatterjee & Dr. Anirban Shaw Science & Physics Learners Python 3 / NumPy / SciPy

1. Core Principles & Theoretical Framework

Dynamical Evolution: Solves differential equations governing kinematics, planetary orbits, electrical circuits, and wave propagation.

Step-by-Step Stepping: Advances continuous state vectors forward in discrete time intervals $\Delta t$ using slope estimates.

Conservation Laws: Evaluates preservation of physical invariants (total energy, angular momentum) across long integration horizons.

2. Mathematical Formulation & Algorithmic Pipeline

Vector initial value problem and numerical state advancement:

$$ \frac{d\mathbf{y}}{dt} = \mathbf{f}(t, \mathbf{y}), \quad \mathbf{y}(t_0) = \mathbf{y}_0 \implies \mathbf{y}_{n+1} = \mathbf{y}_n + \Delta t \cdot \mathbf{\Phi}(t_n, \mathbf{y}_n, \Delta t) $$

3. Numerical Mechanics & Implementation Logic

Coupled Reduction: Convert $n$-th order ODEs into systems of $n$ coupled first-order equations.
Intermediate Slopes: Compute weighted intermediate derivatives ($k_1, k_2, k_3, k_4$ in RK4).
State Update: Advance state variables simultaneously and verify energy/phase invariants.

4. Relevant Computational Practice Exercises

Exercise 1

Solve the differential equation for 1st Order ODE (RK4 Method) over 1000 time steps and plot state coordinates vs time.

Exercise 2

Compare numerical solution stability for small vs large step sizes $h = \Delta t$.

Exercise 3

Plot phase space trajectory demonstrating physical stability or damping attractors.

Standard Literature & References:

John R. Taylor, "Classical Mechanics", University Science Books (2005).
Mark Newman, "Computational Physics", University of Michigan (2013).

Ch. 13.3 1st Order ODE (RK4 Method)