Problem 2. (Runge-Kutta method - J) In this exercise we will study a Runge-Kutta method that is given by k₁ = f(tn, yn) h k2= ftn+ k₁ 3' Yn + 3 ½ 4 ) 1kg = 5 (1₁ + 3 / 1, 3 — — — 1 + kg) k3 | tn h, Yn k4= f (tn+h, yn + k₁ − k₂+ k3) Yn+1 = h : Yn + − (k₁ + 3k2 + 3k3 + k4) 8 a) Present the method in the form of a Butcher tableau. b) Decide the order of the method. c) Implement this method in Python. d) Verify the convergence order numerically. For this you can use the example problem y' = 2ty, y(0) = 1, which has the analytical solution y(t) = et², on the interval [0, 1].
Problem 2. (Runge-Kutta method - J) In this exercise we will study a Runge-Kutta method that is given by k₁ = f(tn, yn) h k2= ftn+ k₁ 3' Yn + 3 ½ 4 ) 1kg = 5 (1₁ + 3 / 1, 3 — — — 1 + kg) k3 | tn h, Yn k4= f (tn+h, yn + k₁ − k₂+ k3) Yn+1 = h : Yn + − (k₁ + 3k2 + 3k3 + k4) 8 a) Present the method in the form of a Butcher tableau. b) Decide the order of the method. c) Implement this method in Python. d) Verify the convergence order numerically. For this you can use the example problem y' = 2ty, y(0) = 1, which has the analytical solution y(t) = et², on the interval [0, 1].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Problem 2. (Runge-Kutta method - J)
In this exercise we will study a Runge-Kutta method that is given by
k₁ = f(tn, yn)
h
k2= ftn+
k₁
3' Yn +
3
½ 4 )
1kg = 5 (1₁ + 3 / 1, 3 — — — 1 + kg)
k3 | tn
h, Yn
k4= f (tn+h, yn + k₁ − k₂+ k3)
Yn+1 =
h
: Yn + − (k₁ + 3k2 + 3k3 + k4)
8
a) Present the method in the form of a Butcher tableau.
b) Decide the order of the method.
c) Implement this method in Python.
d) Verify the convergence order numerically. For this you can use the example problem
y' = 2ty,
y(0) = 1,
which has the analytical solution y(t) = et², on the interval [0, 1].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf310528-7d30-49bd-a043-763f875988f4%2Ffe954d00-4eac-47a4-a605-18f4c7713ac5%2Fekpplo9_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 2. (Runge-Kutta method - J)
In this exercise we will study a Runge-Kutta method that is given by
k₁ = f(tn, yn)
h
k2= ftn+
k₁
3' Yn +
3
½ 4 )
1kg = 5 (1₁ + 3 / 1, 3 — — — 1 + kg)
k3 | tn
h, Yn
k4= f (tn+h, yn + k₁ − k₂+ k3)
Yn+1 =
h
: Yn + − (k₁ + 3k2 + 3k3 + k4)
8
a) Present the method in the form of a Butcher tableau.
b) Decide the order of the method.
c) Implement this method in Python.
d) Verify the convergence order numerically. For this you can use the example problem
y' = 2ty,
y(0) = 1,
which has the analytical solution y(t) = et², on the interval [0, 1].
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