#Module 4:
This module focuses on solution of parabolic PDEs, like the diffusion equation. It introduces for the first time implicit methods and covers both one- and two-dimensional problems.
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Lesson 1 develops a 1D heat-conduction problem and its solution by means of a forward-time/centered-space scheme. It discusses in detail Dirichlet and Neumann boundary conditions, looking at their implementation in code. At the end, it touches on boundary condition and time step limits with explicit schemes.
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Lesson 2 introduces implicit schemes for the first time: it develops the implicit discretization of the 1D heat equation and discusses boundary conditions in detail.
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In lesson we graduate to two dimensions! A 2D heat-conduction problem is described, representing a computer microchip, and is solved with an explicit scheme. The lesson covers boundary conditions in 2D and array-storage decisions.
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Lesson 4 develops the implicit solution of 2D heat conduction, explaining in detail how to construct the coefficient matrix and the various combinations of boundary conditions.
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Lesson 5 is dedicated to the Crank-Nicolson scheme, including a study of spatial and time accuracy and convergence.
##Badge earning Completion of this module in the online course platform can earn the learner the Module 4 badge.
###Description: What does this badge represent? The earner completed Module 4 of the course "Practical Numerical Methods with Python" (a.k.a., numericalmooc).
###Criteria: What needs to be done to earn it? To earn this badge, the learner needs to complete the graded assessment in the course platform including: answering quiz questions about handling boundary conditions, and completing the individual coding assignment on the Gray-Scott model of reaction-diffusion and answering the numeric questions online. Earners should also have completed self-study of the five module lessons, by reading, reflecting on and writing their own version of the codes. This is not directly assessed, but it is assumed. Thus, earners are encouraged to provide evidence of this self-study by giving links to their code repositories or other learning objects they created in the process.
###Evidence: Website (link to original digital content) Desirable: link to the earner's GitHub repository (or equivalent) containing the solution to the "Rocket flight" coding assignment. Optional: link to the earner's GitHub repository (or equivalent) containing other codes, following the lesson.
###Category: Higher education, graduate
###Tags: engineering, computation, higher education, numericalmooc, python, gwu, george washington university, lorena barba, github
###Relevant Links: Is there more information on the web?