Runge-Kutta Method Calculator
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Understanding the Runge-Kutta Method
What is the Runge-Kutta Method?
The Runge-Kutta method is a family of numerical methods for solving ordinary differential equations (ODEs) with high accuracy.
RK4 Formulas:
k₁ = h·f(xₙ, yₙ)
k₂ = h·f(xₙ + h/2, yₙ + k₁/2)
k₃ = h·f(xₙ + h/2, yₙ + k₂/2)
k₄ = h·f(xₙ + h, yₙ + k₃)
yₙ₊₁ = yₙ + (k₁ + 2k₂ + 2k₃ + k₄)/6
Key Concepts
- Order of Accuracy: O(h⁴)
- Local Truncation Error: O(h⁵)
- Stability Analysis
- Step Size Selection
- Error Control
- Adaptive Methods
Advanced Topics
Butcher Tableau
Method coefficients
Embedded Methods
Error estimation
Stiff Problems
Stability regions
Systems of ODEs
Multiple equations
Applications
- Physics: Motion equations
- Engineering: Control systems
- Biology: Population dynamics
- Chemistry: Reaction kinetics
- Economics: Growth models
- Electronics: Circuit analysis