The derivation of k1 and k2 in the context of the second-order Runge-Kutta method

​ ​

involves approximating the solution of an ordinary differential equation (ODE) using

Taylor series expansions. Let's go through the derivation step by step:

Consider the first-order ODE:

dy
dt
​ = f (t, y)

The goal is to find an approximation for y(t + h), where h is the step size.

Taylor Expansion around tn , yn : ​ ​

∣ ∣
dy h2 d2 y
y(t + h) = y(tn ) + h ⋅ ​

dt
​ ​ ​ + 2
⋅ dt2
​ ​ ​ + O(h3 )
∣ (tn ,yn ) ​ ​
∣ (tn ,yn ) ​ ​

Approximate y(t + h) with k1 : ​

If we truncate the series at the first-order term, we get:

y(t + h) ≈ yn + h ⋅ f (tn , yn ) ​ ​ ​

This is our approximation, and we define k1 as: ​

k1 = h ⋅ f (tn , yn )
​ ​ ​

Now, Evaluate k2 : ​

To improve our approximation, we'll include the second-order term. Substitute y(t +
h) into the Taylor expansion:
∣
h2 df
y(t + h) ≈ yn + h ⋅ f (tn , yn ) +​ ​ ​

2
​ ⋅ dt
​ ​

∣ (tn ,yn )
​ ​

Now, let's use the midpoint to estimate f (tn ​ + h2 , yn + ​ ​

k1
2
), which is k2 :
​

​ ​

h k1
k2 = h ⋅ f ( t n +
​ ​ ​,y
2 n
​ + 2
) ​

So, k1 is obtained by approximating the solution using the first-order term, and k2 is
​ ​

obtained by considering the midpoint and incorporating the second-order term into
the Taylor expansion.

You
show the same derivation for ranga kutta fourth order

ChatGPT
Certainly! Let's derive the Runge-Kutta (RK4) method, a fourth-order numerical
method, using Taylor series expansions. Consider a first-order ordinary differential
equation (ODE):

dy
dt
​ = f (t, y)

The RK4 method involves four steps, and we'll derive the expressions for k1 , k2 , k3 , and ​ ​ ​

k4 : ​

Taylor Expansion around tn , yn : ​ ​

∣ ∣ ∣ ∣
dy h2 d2 y h3 d3 y h4 d4 y
y(t + h) = y(tn ) + h ⋅ ​

dt
​ ​ ​ + 2
⋅ dt2
​ ​ ​ + 6
​ ⋅ dt3
​ ​ ​ + 24
​ ⋅ dt4
​ ​ ​ +
∣ (tn ,yn ) ​ ​
∣ (tn ,yn ) ​ ​
∣ (tn ,yn )
​ ​
∣ (tn ,yn )
​ ​

O(h5 )
Approximate y(t + h) with k1 : ​

If we truncate the series at the first-order term, we get:

y(t + h) ≈ yn + h ⋅ f (tn , yn ) ​ ​ ​

This is our approximation, and we define k1 as: ​

k1 = h ⋅ f (tn , yn )
​ ​ ​

Approximate y(t + h) with k2 : ​

Include the second-order term in the Taylor expansion:

∣
h2 df
y(t + h) ≈ yn + h ⋅ f (tn , yn ) + ​ ​ ​

2
​ ⋅ dt
​ ​

∣ (tn ,yn )
​ ​

Define k2 as: ​

k2 = h ⋅ f (tn + h2 , yn +
​ ​ ​ ​
k1
2)
​

Repeat for k3 and k4 : ​ ​

Continue the process for the third- and fourth-order terms:

k3 = h ⋅ f (tn + h2 , yn + k22 )
​ ​ ​ ​
​

k4 = h ⋅ f (tn + h, yn + k3 )
​ ​ ​ ​
Now, the RK4 update formula is:
yn+1 = yn + 16 (k1 + 2k2 + 2k3 + k4 )
​ ​ ​ ​ ​ ​

The RK4 method provides a more accurate approximation of the solution compared to
lower-order methods.