Submitted By ypzhang
NATIONAL UNIVERSITY OF SINGAPORE
ME3291 – NUMERICAL METHODS IN ENGINEERING
(Semester 2 : AY2013/2014)
Time Allowed : 2 Hours
INSTRUCTIONS TO STUDENTS:
Please write your Student Number only. Do not write your name.
This assessment paper contains FOUR (4) questions and comprises FOUR (4) printed pages.
Students are required to answer ALL FOUR (4) questions.
Students should write the answers for each question on a new page.
This is a CLOSED-BOOK ASSESSMENT with authorized materials. Students are allowed to bring two A4 size sheets of notes/formulae written on both sides.
All questions carry equal marks.
The heat conduction equation in 1D is given by
T/ t = b
Here T is the temperature and b is the thermal conductivity.
You are interested to use the DuFort & Frankel discretization scheme to obtain the finite difference equation of the governing equation because you have heard of its inherent stable properties. The DuFort & Frankel scheme is given as:
(Tpq+1 - Tpq-1)/(2 t) = (b / ( x)2) [Tp+1q – (Tpq-1 + Tpq+1) + Tp-1q]. where Tpq = T (p x, q t) is the finite difference representation.
You are interested to use the von Neumann (Fourier) stability analysis to determine if it is inherently stable or otherwise. If otherwise, then you show the criterion for the limit of stability. You may assume that Tpq = q ei ph where is the amplification factor, is a particular spatial
Fourier mode, i is the complex number (-1)0.5, and h x in the analysis.
Determine the (quadratic) equation for
Hence or otherwise, determine the possible range of r ( t/( x)2) so that there is no source or sink in the governing heat conduction equation.
≤ 1.0 since
The one-dimensional wave…...