Suggested Topics for the Mini-Project
You may choose a topic of numerical algorithm nature or a
topic of an application nature.
Topics of
Numerical Algorithm Nature:
- (NA1) Multigrid Method: Develop
a MATLAB code which solve a linear 4th order BVP
using two-grid method. Start by reading section 7.3 and iterative refinement
in 7.5. You may consult the slides in
here or
A
Multigrid Tutorial, Second Edition.
- (NA2) Locating Saddle Point: Develop
a MATLAB code which which
finds all saddle points of a function in two variables f(x, y).
- (NA3) Eigenvalues of a Matrix Develop a MATLAB code which
Calculate all eigenvalues of a matrix A_(n×n) using LU-factorization. Start by reading sec 6.5
then search for QR method.
- (NA4) Domain Decomposition: Develop a MATLAB code which
solve a nonlinear ode BVP using domain decomposition. start by learn how to
use parfor Matlab command.
- (NA5) Newton-type methods with cubic convergence: Develop a MATLAB code which
finds roots of a function. Start by reading the first three pages of
the this paper "Homeier, H. H. H. "On Newton-type methods with cubic
convergence." Journal of computational and applied mathematics 176.2 (2005):
425-432."
-
(NA6) Node ordering Study: Use FDM to descritize Laplace
equation then Develop a MATLAB code which
assemble the matrix with different strategy of ordering then watch cpu time
of the linear solver. Start by reading sec 11.3, 7.3 and search Red-black
ordering.
-
(NA7) Chebfun for Darcy's equation: Use
Chebfun to solve 1D Darcy-Forchheimer
equation. Start by getting the equation from “A Block-Centered
Finite Difference Method For The Darcy–Forchheimer Model” by Hongxing Rui and Hao Pan, SIAM J. Numer. Anal. vol. 50, no. 5, pp. 2612–2631" then solve it by
chebfun. some links
https://en.wikipedia.org/wiki/Chebfun http://www.chebfun.org/docs/guide/guide10.html
-
(NA8) Numerical Comparison Study for 4th order ode: Use
Chebfun to solve 4th order BVP and compare the solution with FDM. Start
by reading sec 11.3. some links
https://en.wikipedia.org/wiki/Chebfun http://www.chebfun.org/docs/guide/guide10.html
-
(NA10) Bisection method and cubic spline: Develop an
algorith for finding roots of equations using Bisection method and cubic
spline.
- (NA12) Find a Minimum of a function: Prof. Mike Powell
developed a Fortran code LINCOA. It is software for minimization without
derivatives. Contact Prof. Mike to get the code and his permission to
convert it to Matlab and use to solve a minimization problem.
- (NA13) Preconditioning Technique: Use Finite difference
methods to solve 2ed or 4th order ode using two different scheme one with
low order and the second with high order. Use the low order scheme matrix as
a preconditioner to solve the linear system generated by the high order
scheme. Start by reading sec 11.3 and search for preconditioning linear
system or see these
slides.
- (NA14) Toeplitz matrices: search for circulant and |Toeplitz
matrices and BCCB and BTTB matrices. Then suggent and develop Matlab code
which solve linear system whose coefficient matrix is T*T where T is BTTB.
Start by reading sec 7.3.
Topics of
Application Nature:
§
(APP2)
Heat equation for Arabian peninsula: Use your data in cubic spline
section to solve the heat equation for a plate where the plate is the Arabian
peninsula. use pdetool box to solve and visulize.
§
(APP4)
Choose a Case Study from Chapter 8 in Numerical Methods for Engineers”, Steven
C. Chapra and Raymond P. Canale. (6th Edition):
All problems in this section are finding a root of a single equation.
You have to consider the system of non-linear equations.
8.1 Ideal and Nonideal
Gas Laws (Chemical/Bio Engineering) 202
8.2 Greenhouse Gases and Rainwater
(Civil/Environmental Engineering) 205
8.3 Design of an Electric Circuit
(Electrical Engineering) 207
8.4 Pipe Friction
(Mechanical/Aerospace Engineering) 209
§
(APP5)
Choose a Case Study from Chapter 12 Numerical Methods for Engineers”, Steven C.
Chapra and Raymond P. Canale. (6th Edition): All problems in this section
required solving linear system of equations. You have to consider the cases of
solving linear system of nonlinear equations.
12.1
Steady-State Analysis of a System of Reactors (Chemical/BioEngineering)
315
12.2
Analysis of a Statically Determinate Truss (Civil/Environmental Engineering)
318
12.3
Currents and Voltages in Resistor Circuits (Electrical Engineering) 322
12.4Spring-Mass
Systems (Mechanical/Aerospace Engineering) 324
§
(APP6)
Google Page rank: Google Page
rank problems: Use PageRank Algorithm to Rank KFUPM Websites. Explore methods
of calculating the eigenvector associated with lambda 1. https://uk.mathworks.com/help/matlab/examples/use-page-rank-algorithm-to-rank-websites.html