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Université
Paris 13
MACS 1 - Modeling and case studies:
differential equations
Professors : Caroline Japhet, Pascal Omnes
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Many physical
phenomena can be formulated as differential equations, for
example a simple
pendulum (see also here),
a predator-prey model,
... Most of these equations however can not be solved exactly by
analytical tools. Their solution has to be approximated using
numerical methods. Among these methods we will study :
- explicit and implicit Euler, Crank-Nicolson, and BDF schemes
-
first
and second orders linear differential
equations
- nonlinear differential
equations, requiring
the resolution of a nonlinear algebraic equation at each
time step by fixed-point
or Newton methods
- differential
systems
- finite
difference method
- how
to write the corresponding codes in C language, using
separate compilation and Makefile, dynamic memory
allocation, and function pointers
- how
to visualize the numerical solutions as functions of
time, and the error schemes, using gnuplot.
After the course you will be able, from a given problem and
method, to write the corresponding codes in C languageto recover the theoretical results
(consistency, stability,...) studied in class, compare
differentnumerical methods and choose the "best one" to get
an accurate solution, close to the physics with a minimum
computational time andstorage., and to
validate these codes by finding the theoretical
results (convergence to the exact solution if known, scheme
error, stability, ...) and the physics studied in class.
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