Changes between Version 33 and Version 34 of u/erica/PoissonSolver

Timestamp:

08/20/13 10:40:31 (11 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/PoissonSolver

@@ -29 +29 @@
 The solutions to both Poisson's and Laplace's equations satisfy the Uniqueness theorem (see Griffith's E&M textbook). That is, if the solution satisfies the aforementioned 2 conditions, it is the ''unique'' solution to the equation. 
 
-As is, Poisson's equation is set up to solve for f (be it charge density, or matter density, etc.) given some potential (e.g. electrical or gravitational, etc.). However, in most situations we have the opposite information about a system -- given the density, we seek the potential. This requires some numerical techniques for solving this 2nd order differential equation, especially for those systems that do not admit closed-form solutions.
+As is, Poisson's equation is set up to solve for f (be it charge density, or matter density, etc.) given some u (e.g. potential- either electrical or gravitational, etc.). However, in most situations we have the opposite information about a system -- given the density, we seek the potential. This requires some numerical techniques for solving this 2nd order differential equation, especially for those systems that do not admit closed-form solutions.
 
 = Equation Discretization = 
@@ -88 +88 @@
 = Poisson Equation Physical Meaning = 
 
-While the interpretation of Laplace's equation and the harmonic functions are nice, the same does not seem to apply for the more general Poisson's equation. Instead, the solutions to Poisson's equation are called Green's functions, and they are stitched together out of the solutions to the homogenous part of the PDE (namely, Laplace's equation. Here are some references:
+While the interpretation of Laplace's equation and the harmonic functions are nice, the same does not seem to apply for the more general Poisson's equation. Instead, the solutions to Poisson's equation are called Green's functions, and they are stitched together out of the solutions to the homogenous part of the PDE (namely, Laplace's equation). Here are some references:
+
+[http://www.math.osu.edu/~gerlach.1/math/BVtypset/node87.html constructing Green's functions]
+
+[http://farside.ph.utexas.edu/teaching/em/lectures/node31.html related to Poisson's equation]
 
 = Boundary Conditions =
@@ -117 +121 @@
 This also has a closed-form solution which I then used to check the code. It performed correctly.
 
-Here is the initial condition:
+Here is the forcing function:
 
 [[Image(simpleInit.png, 40%)]]
@@ -131 +135 @@
 Lastly, I used a more complicated forcing function (refer to the reference). It also performed well. 
 
-Here is the initial condition:
+Here is the forcing function:
 
 [[Image(complexInit.png, 40%)]]
@@ -139 +143 @@
 [[Image(complexFinal.png, 40%)]]
 
-Attached to this page is the .out file for my program which shows the number of iterations until convergence for the different mesh sizes, and the max error of the final iteration.
+'''Attached to this page is the .out file''' for my program which shows the number of iterations until convergence for the different mesh sizes, and the max error of the final iteration.
 
 
@@ -150 +154 @@
 
 = References =
+
+Here are some of my electronic references:
+
+- [https://en.wikipedia.org/wiki/Order_of_accuracy Order of accuracy]
+
+- [http://farside.ph.utexas.edu/teaching/em/lectures/node31.html Green's functions and Poisson's equation]
+
+- [http://www.math.osu.edu/~gerlach.1/math/BVtypset/node87.html Constructing Green's functions]
+
+- [http://www.rsmas.miami.edu/personal/miskandarani/Courses/MSC321/Projects/prjpoisson.pdf Exercises I followed for my Poisson solver]
+
+
+- [http://hipacc.ucsc.edu/html/HIPACCLectures/lecture_source.pdf Poisson solver info]
+
+- [http://geo.mff.cuni.cz/~lh/GUCAS/PDEwPGI3.pdf more PS info]
+
+- [http://people.sc.fsu.edu/~jburkardt/presentations/jacobi_poisson_1d.pdf more PS info]
+
+- [http://jupiter.ethz.ch/~pjt/FORTRAN/Class6.pdf more PS info]
+
+(The .pdf's are also attached to this page, in case they get taken down)
+
+As far as books, I referenced:
+
+- Numerical Recipes in Fortran - chapter 19 on PDE's
+
+- Leveque's Finite Differences
+