Runge–Kutta Methods of Order 2
We have seen that the development of the Taylor method relies on relation (7.7):
y(tₙ₊₁) = y(tₙ) + h f(tₙ, y(tₙ))
+ (h²/2) [ ∂f/∂t (tₙ, y(tₙ)) + (∂f/∂y)(tₙ, y(tₙ)) f(tₙ, y(tₙ)) ] + O(h³)
The objective is to replace this expression by another relation of the same precision order (O(h³)), but without requiring partial derivatives. We propose the general form:
y(tₙ₊₁) = y(tₙ) + a₁ h f(tₙ, y(tₙ))
+ a₂ h f(tₙ + a₃ h , y(tₙ) + a₄ h)
The coefficients a₁, a₂, a₃, a₄ must be chosen so that both expressions have truncation error in O(h³). To achieve this, we use the two-variable Taylor expansion (see Section 1.6.2):
f(tₙ + a₃ h , y(tₙ) + a₄ h) = f(tₙ, y(tₙ))
+ a₃ h (∂f/∂t)(tₙ, y(tₙ)) + a₄ h (∂f/∂y)(tₙ, y(tₙ))
Substituting this expansion in the RK2 formula gives:
y(tₙ₊₁) = y(tₙ) + (a₁ + a₂) h f(tₙ, y(tₙ))
+ a₂ a₃ h² (∂f/∂t)(tₙ, y(tₙ))
+ a₂ a₄ h² (∂f/∂y)(tₙ, y(tₙ)) + O(h³)
We now compare term by term with the Taylor expansion to determine the coefficients:
Matching Coefficients
• Coefficient of f(tₙ, y(tₙ)):
h = (a₁ + a₂) h
• Coefficient of ∂f/∂t:
h²/2 = a₂ a₃ h²
• Coefficient of ∂f/∂y:
(h²/2) f(tₙ, y(tₙ)) = a₂ a₄ h²
This yields the nonlinear system:
{
1 = a₁ + a₂
1/2 = a₂ a₃
1/2 = a₂ a₄
}
This system contains three equations but four unknowns: it is underdetermined. Therefore, the Runge–Kutta formulation admits multiple valid choices of coefficients. This flexibility leads to several variants of second-order Runge–Kutta schemes.
The most commonly used choice of parameters will be presented in the next section.