I'm trying to show that RK4 has local truncation error of order 4.

My book's definition of local truncation error is:

[y_(i+1) - y_i] / h - g(t_i, y_i)

where the method has the form u_(i+1) = u_i + h*g(t_i, u_i)

So for RK4 I have g given. I tried to taylor expand y_(i+1) about
y_i, but I don't get nice cancellation (as you would for Euler's
method). Is this the wrong approach?

I assume I could also write out taylor's method of order 4, and expand
in 2-dimensions, but this seems very painful to expand up to order 4.
If I were to do this, should I expand each of the following f's about
(t_i, u_i)? Or is there another point I should expand about [or
yes, clearly
several different points]?

k1 = h*f(t_i, u_i)
k2 = h*f(t_i + h/2, u_i + k1/2)
k3 = h*f(t_i + h/2, u_i + k2/2)
k4 = h*f(t_(i+1), u_i + k3)

I'm trying to show that RK4 has local truncation error of order 4.

My book's definition of local truncation error is:

[y_(i+1) - y_i] / h - g(t_i, y_i)

where the method has the form u_(i+1) = u_i + h*g(t_i, u_i)

So for RK4 I have g given. I tried to taylor expand y_(i+1) about
y_i, but I don't get nice cancellation (as you would for Euler's
method). Is this the wrong approach?

I assume I could also write out taylor's method of order 4, and expand
in 2-dimensions, but this seems very painful to expand up to order 4.
If I were to do this, should I expand each of the following f's about
(t_i, u_i)? Or is there another point I should expand about [or
several different points]?

k1 = h*f(t_i, u_i)
k2 = h*f(t_i + h/2, u_i + k1/2)
k3 = h*f(t_i + h/2, u_i + k2/2)
k4 = h*f(t_(i+1), u_i + k3)


There should be quite an easier method for showing that.:::

See this method:


Notation

f_n = f(x_n, t_n)

The general k-step method can be written as

x_[n+] = phi(f_[n+1], f_n, ...., f_[n-k+1], x_n, ... x_[n-k+1], h)


K-step methods are linear dependend on f_[n+1], f_n, ...., f_[n-k+1],
x_n, ... x_[n-k+1], and can therefore be written as (rewriting done)

sum_j=0^k (alpha_j * x_[n+j]) = h * sum_j=0^k (beta_j f_[n+j])



Let c_i be given by:

c_0 = sum_j=0^k (alpha_j)

c_1 = sum_j=0^k (j*alpha_j - beta_j)

c_i = sum_j=1^k( 1/(i!) * j^i * alpha_j - 1/((i-1)!) * j^(i-1) * beta_j


Then the order p of the method is given by

c_0 = c_1 = ... = c_p = 0, c_[p+1] != 0


It's a lot easier this way...

Hope I didn't do any typo's... Try it out!

I don't know the name of this method, but it's only valid for linear
k-step methods.
You have to rewrite the method, so you don't have evaluation's in ½
time-steps. This should be possible as far as I can see.

I'm trying to show that RK4 has local truncation error of order 4.

My book's definition of local truncation error is:

[y_(i+1) - y_i] / h - g(t_i, y_i)

where the method has the form u_(i+1) = u_i + h*g(t_i, u_i)

So for RK4 I have g given. I tried to taylor expand y_(i+1) about
y_i, but I don't get nice cancellation (as you would for Euler's
method). Is this the wrong approach?

I assume I could also write out taylor's method of order 4, and expand
in 2-dimensions, but this seems very painful to expand up to order 4.
If I were to do this, should I expand each of the following f's about
(t_i, u_i)? Or is there another point I should expand about [or
several different points]?

k1 = h*f(t_i, u_i)
k2 = h*f(t_i + h/2, u_i + k1/2)
k3 = h*f(t_i + h/2, u_i + k2/2)
k4 = h*f(t_(i+1), u_i + k3)

In article <45ec4ab2$0$90266$14726298@news.sunsite.dk>,
Preben <64bitNONOSPAMno@mailme.dk> writes:
something below confusing linear multistep methods which an
Runge-Kutta method which is highly nonlinear.
the consistency conditions for linear multistep meethods are indeed
much easier.
unfortunately here it is not that easy

I'm trying to show that RK4 has local truncation error of order 4.

My book's definition of local truncation error is:

[y_(i+1) - y_i] / h - g(t_i, y_i)

where the method has the form u_(i+1) = u_i + h*g(t_i, u_i)

So for RK4 I have g given. I tried to taylor expand y_(i+1) about
y_i, but I don't get nice cancellation (as you would for Euler's
method). Is this the wrong approach?

I assume I could also write out taylor's method of order 4, and expand
in 2-dimensions, but this seems very painful to expand up to order 4.
If I were to do this, should I expand each of the following f's about
(t_i, u_i)? Or is there another point I should expand about [or
several different points]?

k1 = h*f(t_i, u_i)
k2 = h*f(t_i + h/2, u_i + k1/2)
k3 = h*f(t_i + h/2, u_i + k2/2)
k4 = h*f(t_(i+1), u_i + k3)

Order := 5;
de:= D(y)(t) = f(t,y(t)):