Physics 212, 2019: Lecture 9 - Revision history

Ilya: /* Your work */

2019-02-14T14:17:57Z

‎Your work

Ilya: /* Scripts for this Lecture */

2019-02-13T14:50:50Z

‎Scripts for this Lecture

Ilya: /* Runge-Kutta 2 integration */

2019-02-13T14:29:23Z

‎Runge-Kutta 2 integration

Ilya: /* Runge-Kutta 2 integration */

2019-02-13T14:12:22Z

‎Runge-Kutta 2 integration

Ilya: Created page with "{{PHYS212-2019}} ==Runge-Kutta 2 integration== ===Scripts for this Lecture=== :: GrowthFunctions -- module defining different growth func..."

2019-02-13T14:04:28Z

Created page with "{{PHYS212-2019}} ==Runge-Kutta 2 integration== ===Scripts for this Lecture=== :: GrowthFunctions -- module defining different growth func..."

New page

{{PHYS212-2019}}

==Runge-Kutta 2 integration==

===Scripts for this Lecture===
::[[media:growthfunctions2019.txt | GrowthFunctions]] -- module defining different growth functions.
::[[media:Integrators2019.txt | Integrators]] -- module defining different integrators (Euler and Runge-Kutta, which we will use during the next lecture).
::[[media:moduleTwo2019.txt | Module 2]] -- catch-all scripts I used during the class.

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 ===Your work===
 ;Problem 1: Use a different differential equation from those provided to show that Runge-Kutta 2 has error <math>O(dt^2)</math>, and the scaling of the error is not a scaling due to the equation being solved, but largely due to the methods used to solve it. Do this in a Jupyter notebook.
 ===Scripts for this Lecture===

← Older revision		Revision as of 14:50, 13 February 2019
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	::[[media:Integrators2019.txt \| Integrators]] -- module defining different integrators (Euler and Runge-Kutta, which we will use during the next lecture).		::[[media:Integrators2019.txt \| Integrators]] -- module defining different integrators (Euler and Runge-Kutta, which we will use during the next lecture).
	::[[media:moduleTwo2019.txt \| Module 2]] -- catch-all scripts I used during the class.		::[[media:moduleTwo2019.txt \| Module 2]] -- catch-all scripts I used during the class.
		+	::[[media:IntegrateODE.txt \| Module 2 as a notebook]] -- integration as a notebook.

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 ===Your work===
-;Problem 1: Use a different differential equation from those provided to show that Runge-Kutta 2 has error <math>O(dt^2)</math>, and the scaling of the error is not a scaling due to the equation being solved, but largely due to the methods used to solve it.
+;Problem 1: Use a different differential equation from those provided to show that Runge-Kutta 2 has error <math>O(dt^2)</math>, and the scaling of the error is not a scaling due to the equation being solved, but largely due to the methods used to solve it. Do this in a Jupyter notebook.
-;Problem2: Write a system of differential equations that correspond to a harmonic oscillator (linear pendulum) and solve it using a multi-dimensional generalization of the RK2 algorithm.
+;Problem2: Write a system of differential equations that correspond to a harmonic oscillator (linear pendulum) and solve it using a multi-dimensional generalization of the RK2 algorithm. Do this in a Jupyter notebook.
 ===Scripts for this Lecture===

← Older revision		Revision as of 14:12, 13 February 2019
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	==Runge-Kutta 2 integration==		==Runge-Kutta 2 integration==
		+	How do we solve systems on nonlinear ODEs in the first place? The Euler method, which we introduced before, is good. But how accurate is it? We will talk about <math>O(dt^n)</math> notation, and we will show that the Euler method is <math>O(dt)</math>. We will verify this by using the scripts that I have uploaded below. We will run the simple Euler method of solving differential equations on the malthusian grows problem with different <math>dt</math>, and show by plotting the error at the end that the error goes down in proportion to <math>dt</math>.

		+	We then will talk about Runge-Kutta methods of solving differential equations. I will add additional notes later on, but for now please use these lecture notes http://web.mit.edu/10.001/Web/Course_Notes/Differential_Equations_Notes/node5.html . Additionally, we discussed how RK2 method is a prediction-correction method.
		+
		+	===Your work===
		+	;Problem 1: Use a different differential equation from those provided to show that Runge-Kutta 2 has error <math>O(dt^2)</math>, and the scaling of the error is not a scaling due to the equation being solved, but largely due to the methods used to solve it.
		+
		+	;Problem2: Write a system of differential equations that correspond to a harmonic oscillator (linear pendulum) and solve it using a multi-dimensional generalization of the RK2 algorithm.

	===Scripts for this Lecture===		===Scripts for this Lecture===