The Runge-Kutta (RK) method is a method of numerically integrating ordinary differential equations (ODEs). An estimation is found by taking the midpoint of an interval to cancel out lower-order terms. The power of this method (or family of methods) is that there are different orders of these methods. An arbitrarily large-ordered RK method can be […]

For a long time, baseball lovers have been interested in the physics of a baseball’s flight. One particularly big question is how the seams on a baseball affect the way it flies from the pitcher to the home plate. A mechanical engineer, LeRoy Alaways has been pursuing the answer to this question. Identifying factors such […]

As we learned in class, the basic principle of the Monte Carlo method is to randomly sample data points and compute their results. These methods are especially useful when we cannot find a definite algorithm for the process, such as finding the volume of an irregular shape.
Thus a foundational tool in any Monte Carlo algorithm […]

Completing the series of blog posts on the histories of mathematicians whose theorems we have learned, here are some interesting historical tidbits on Carl David Tolmé Runge and Martin Wilhelm Kutta of the (famously painful to derive as we are all aware of after our homework problem) Runge-Kutta Method.
 Carl Runge: 

Though originally enrolled at the University of Munich […]

I thought that the complex explosions displayed in the assignment looked really cool and decided to investigate further into how these were rendered. I looked up the paper where these images came from and found quite an interesting paper on rendering large explosions.
The paper was concerned with how 3d explosions were being rendered […]

We have seen in class that we can use Monte Carlo methods to estimate the integral of a function of a tricky domain. In general, Monte Carlo methods can be applied to any number of problems in which it is difficult to construct a deterministic method to solve the problem. One […]

Computing the first few (150 or so) factorials is a fairly trivial operation. The classic recursive implementation is well known and simple to understand.
fact (n):
if(n<=1)
return 1
return n * fact(n-1)
end
What happens then when we use this to try and calculate a large factorial (say >200). Depending on the system this code is running on […]

So far we have studied Newton’s method for root finding, Newton’s method for polynomial interpolation, and the Newton-Cotes methods for numerical integration.  You’ve probably also heard of Newton’s laws of planetary motion, Newton’s three laws of motion, Newton’s  and the SI unit of force, the Newton.
We owe a lot to Sir Isaac Newton.  Among these […]

Monte Carlo is an extremely useful technique for solving many-bodied quantum systems.
The basis for quantum mechanics is the Schroedinger equation which is a second-order differential equation that relates the wavefunction of a system to its total energy: [1]

Among the problems in solving this equation is the fact that upon adding multiple particles to the system […]

http://www.mathworks.com/company/newsletters/news_notes/clevescorner/may03_cleve.html
In class, we’ve discussed some numerical methods for solving initial value problems (Euler’s method being one of them). One thing mentioned but unexplored in lecture was the concept of stiff differential equations. An equation is said to be stiff if some component of the equation varies slowly while another varies quite rapidly. The book (Cheney […]