2013-Gonze, Didier - Numerical methods for Ordinary Differential Equations. Notes.
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Gonze, Didier. 2013. Numerical methods for Ordinary Differential Equations. Notes. 15 pp.
See http://homepages.ulb.ac.be/~dgonze/TEACHING/numerics.pdf. Accessed 8 September 2017.
The paper begins,
“Differential equations can describe nearly all systems undergoing change. They are widespread in physics, engineering, economics, social science, but also in biology. Many mathematicians have studied the nature of these equations and many complicated systems can be described quite precisely with mathematical expressions. However, many systems involving differential equations may be complex (nonlinear) and can count several coupled equations. Thus, a purely mathematical analysis is often not possible. Computer simulations and numerical approximations are then needed.
“The techniques for solving differential equations based on numerical approximations were developed long before programmable computers existed. It was common to see equations solved in rooms of people working on mechanical calculators. As computers have increased
in speed and decreased in cost, increasingly complex systems of differential equations can be solved on a common computer. Currently, your laptop could compute the long term trajectories of about 1 million interacting molecules with relative ease, a problem that was inaccessible to the fastest supercomputers some 10 years ago.”
This article offers rich examples of these methods and reflects upon the question, “How do I know my answer is right?’”
Keywords: numerical methods, differential equation, Euler Method, error estimate, Backward Euler Method, Huen Algorithm, Runge-Kutt Algorithm, stiff systems
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