4th Order Runge-Kutta method and over/under estimates

[dwdoyle8854]

Homework Statement

"Use Excel to approximate dF/dt=-0.1F+70, F(0)=0 to generate approximations for F at t=1,2 and 4 using step size 0.1. Explain whether these approximation are greater than or less than the exact values. Determine whether the shape of the solution curve is increasing, decreasing, concave up or concave down based on the data alone. Explain."

Homework Equations

ynext = ynow + (1/6)(k1 +2k2 + 2k3 + k4)
k1= Δx*f'(xnow, ynow)
k2= Δx*f'(xnow+.5Δx,ynow +.5k1)
k3= Δx*f'(xnow+.5Δx,ynow +.5k2)
k4= Δx*f'(xnow+Δx,ynow+k3)

I found the exact solution to be F(t)=700-700*exp(-.1t)

The Attempt at a Solution

i've attached my excel file.

Since Runge-Kutta is al inear technique and I observed all the slopes in RK to be decreasing I predicted that the method would give an over estimate. However, comparing my estimate with the exact solution shows that I infact get an under-approximation. I am completely lost as to why this occurs and am looking for some explanation. I do not feel i am able to say anything about the concavity given my results that the decreasing rates (f'1, f'2 et cetera on my excel) gave an underestimation.

futher, how exactly does putting weight on the k2 and k3 terms affect the approximation?




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[D H]

RK4 is not a linear technique. Where did you get the idea that it is?

 

[dwdoyle8854]

doesnt it assume that rate of change is constant over an interval of time? that to me says linear. but nonetheless still don't know what to say about the concavity or the reason why I got an underestimation.

 

[D H]

doesnt it assume that rate of change is constant over an interval of time? that to me says linear.

No. That's the whole point of evaluating the derivative four times during one step. The "4" in RK4 is short for fourth order. RK4 essentially comes up with a fourth order polynomial for each step.

 

[dwdoyle8854]

okay, so since its a polynomial, is there anything that can be said about whether we expect it to over/under approximate?

 

[D H]

It's not some random 4th order polynomial. It's a very specific 4th order polynomial. Which is it? What does that tell you about the error? This is homework, so at this stage I'm leaving the rest up to you.

 

[dwdoyle8854]

to be honest, i have no idea what you are hinting at with the whole 4th order polynomial thing. I don't know what that has to do with error, or what precisely you mean by a 4th order polynomial.

this is more of a discovery project than a homework. Its graded sure, but its purpose is to explore.