Lecture 5: Numerical Methods for Solving Ordinary Differential Equations

Overview of the Module

• Focus on numerical methods for solving ordinary differential equations
• Initial value problems discussed
• Covered Runge-Kutta methods: RK2 and RK4

Runge-Kutta Methods

RK2 Method

• Formula:
  [ y_{i+1} = y_i + h \times \text{slope} ]
  where [ ext{slope} = w_1 k_1 + w_2 k_2 ]
• Weights: ( w_1, w_2 ) used to compute slopes ( k_1, k_2 )_

RK4 Method

• Formula:
  [ y_{i+1} = y_i + h \times \left( w_1 k_1 + w_2 k_2 + w_3 k_3 + w_4 k_4 \right) ]
• Calculate slopes:
  • ( k_1 = f(y_i, t_i) )
  • ( k_2 = f(y_i + \frac{h}{2} k_1, t_i + \frac{h}{2}) )
  • ( k_3 = f(y_i + \frac{h}{2} k_2, t_i + \frac{h}{2}) )
  • ( k_4 = f(y_i + h k_3, t_i + h) )
• Weights: ( w_1 = w_2 = \frac{1}{2}, w_3 = \frac{1}{2}, w_4 = 1 )_

Error Analysis

• RK2 method: order of error ( O(h^2) )
• RK4 method: order of error ( O(h^4) )
• RK-GIL method: popular RK4 method for non-adaptive solutions

Predictor-Corrector Methods

Ewan's Method

• Uses RK2 as a base
• Predictor step: Calculates predicted value using slopes
• Corrector step: Utilizes trapezoidal rule to refine the result
• Recursive application helps improve accuracy

Stability Issues

• Stability refers to the behavior of numerical methods as step size ( h ) changes
• Increasing ( h ) can lead to inaccuracies and instability
• Example: Observations of concentration values becoming negative at larger step sizes
• Conclusion: Implicit methods are generally more stable than explicit methods

Next Lecture

• Delve into stability issues in more detail
• Discuss conditions for stability of the explicit Euler's method
• Focus on specific equations and their stability conditions.