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Course Overview
Many numerical simulations of physical phenomenon require the computation of fluid motion. From the traditional
simulations of heat and mass transfer, to biological simulations of living systems, to atmospheric simulations,
the calculation of fluid motion is a fundamental component. Fortunately, even though there is a broad range of
applications, the set of equations used to describe fluid motion is often the same --- the time dependent incompressible
Navier-Stokes equations. This course is concerned with numerical methods that are used to generate solutions to
these ubiquitous equations.
The range of numerical techniques to solve the Navier-Stokes equations is almost as broad as the equations applicability.
However, there is a common set of ideas that underlay the numerical methods, and the goal of the course is to present
these fundamental ideas and show how, after a choice of approximation technique is made, numerical solution procedures
are constructed. To be concrete, we will be experimenting with these ideas in a finite difference context. After
taking this class, a student should be able to understand, work with, and construct numerical simulations that
are based on other approximation methods (e.g. spectral or finite element discretizations).
Requirements:
A course in the numerical solution of PDE's. A fluids course is not essential since we are primarily concerned
with the equations from a mathematical perspective and the requisite background material in fluid mechanics will
be covered in the course. (For mathematics students, this course might as well be considered as "computation
of time dependent divergence free velocity fields".)
Homework :
There will be weekly assignments, each of which involves programming. Students will build their programs by combining
components from C++ class libraries. By the end of the quarter students will have constructed a Navier-Stokes solver
in both the vorticity-stream function formulation and the pressure-velocity formulation. In addition to leaning
about procedures for solving the Navier-Stokes equations, the homework assignments will also serve to familiarize
the students with current computational practices associated with solving multi-dimensional partial differential
equations.
Textbook: Computational Methods for Fluid Flow by R. Peyret and T. D. Taylor. The text is out of print so photo-copied
versions (legal ones) are available for purchase at the UCLA bookstore.
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