Methods for Solving 1st Order Differential Equations

Consider the following discrete methods of solution for the first order linear differential equation:

dy/dx = f(x,y)

Euler Method: Uses slope at beginning of interval to get next point.

y_n+1 = y_n + f(x_n,y_n)Dx

Verlet, Modified Euler, or Central Difference Method: Uses average of slope at the endpoints of the interval.

y_n+1 = y_n + [f(x_n,y_n) + f(x_n+1, y_n + f(x_n,y_n)Dx)] Dx / 2

An equivalent expression for the derivative is

dy(x)/dx)_n = (y_n+1 - y_n-1) / 2Dx

Second Order Runge-Kutta Method: Predict midpoint of interval by Euler method and then use the slope at the midpoint as the slope over the entire interval.

y_n+1 = y_n + k₂

with

k₂ = f(x_n + Dx/2, y_n + k₁/2) Dx

k₁ = f(x_n,y_n) Dx

Fourth Order Runge-Kutta Method: Extension of 2nd order RK to include a quadratic term in the predictions, not just a linear term.

y_n+1 = y_n + [k₁ + 2*k₂ + 2*k₃ + k₄] Dx/6

with

k₁ = f(x_n, y_n)

k₂ = f(x_n + Dx/2, y_n + k₁ Dx/2)

k₃ = f(x_n + Dx/2, y_n + k₂ Dx/2)

k₄= f(x_n + Dx, y_n + k₃ Dx)

The 4th order RK method is more accurate than the Verlet method although it does require 4 evaluations of f(x,y) compared to 2 for the Verlet.