Problem 1a: Elliptical PDE
The cylinder had a radius and height of 1, while its base and sides were kept at a
ground potential (u=0) and the top surface was at u=1. To compute approximations using
the method of finite differences, discretization was performed to represent the geometry
as a continuous system of nodes. By introducing an index notation and constructing the
finite difference approximations, a system of interlinked simultaneous linear algebraic
equations was created. This solution of algebraic equations was solved using the iterative
methods: Jacobian, Gauss-Seidel, and SOR. It can be seen by the text files associated
with each methods fortran sheet that the SOR method solves the system with the same
accuracy much faster. The text files for each method were then plotted using GNUPLOT.
As shown, the plots represent the solutions for the problem which was solved in
homework 4 and part b of homework 5. Taking a cross section of the cylinder at a
constant height, the solution (the potential) increases to a maximum (u=1), at r=0, and
decreases to u=0, at r=1. This curve should be the Bessel J function. Lines of potential
therefore represent a minimum at r=0,z=0 since the potential goes from a value of 1 at r=1 to a minimum at r=0 back to a potential of 1 at r=1. This represents the sinh function, as
shown by the contour lines in the plots.
Method Type
Jacobian
Gauss-Seidel
SOR
Number of
Iteratives
422
257
87
Jacobian Method Plot
Problem 1a: Elliptical PDE
Gauss-Seidel Method Plot
SOR Method Plot