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Ch. 13.1 1st Order ODE (RK2 Method)

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Program 1

1st Order ODE (RK2 Method)

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Click "Run Code" to execute script and render figures.

                                

                            


                            
                                                        

                                

                                                                        

                                        
 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 2nd order (RK2) method to calculate the velocity of the object over a time period.

                                    

                                    
                                                                        

                                        
 Theoretical Background & Computational Mechanics

                                        
        



RK2 Method



    In general, the RK2 method for solving an ODE of the form 
    \( \frac{dy}{dt} = f(t, y) \) over a time interval \([t_0, t_{\text{end}}]\)
    with initial condition \( y(t_0) = y_0 \) is implemented as follows:




    
    
  1. Compute \( k_1 \), an estimate of the slope at the beginning of the interval:
        \[
        k_1 = f(t, y)
        \]
    
    2. 
    
  2. Compute \( k_2 \), an estimate of the slope at the end of the interval using \( k_1 \):
        \[
        k_2 = f\left(t + h, y + h \cdot k_1\right)
        \]
    
    3. 
    
  3. Update the value of \( y \) for the next step \( y_{\text{next}} \) by averaging these slopes:
        \[
        y_{\text{next}} = y + \frac{h}{2} \cdot (k_1 + k_2)
        \]
    
    4. 


                                    

                                                                    

                            

                                                    

                    

                

                            
            
            
            
            
            

                

                    

                         Theoretical Principles & Computational Physics Guide
                    

                    

                        1st Order ODE (RK2 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 (RK2 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).
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  • Mark Newman, "Computational Physics", University of Michigan (2013).
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