Assn1_2011 - GLY 6826 Assignment 1 Finite-difference...

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GLY 6826 Assignment 1 Finite-difference modeling using a spreadsheet 60 points Due Sept 7, 2011 In this assignment, you will work through some basics of finite-difference numerical modeling and start modeling using a spreadsheet program. Purpose : To provide insight on how finite-difference approximations work and to introduce basic concepts of iterations and convergence criteria. Please submit your answers online through the Sakai web site. Include an Excel workbook with your spreadsheets. Part I. Finite Difference Approximation of 1-D steady state flow (25 pts) Consider a one dimensional, homogeneous “aquifer” with steady-state fluid flow. 1. (1 pt) What is the appropriate governing equation for fluid flow? To work up the finite difference approximation for the hydraulic head at a point “i” (so we’ll refer to the hydraulic head as h i ), consider 2 locations in the aquifer that are equally spaced on either side of i (call them i-1 and i+1) and the distance between each (i and i+1 and i and i-1) is x. The first derivative of h with respect to distance, dh/dx, is replaced by the difference, h/ x . 2. A. (1 pt) Write an expression for h/ x at point i-1/2, located half way between i- 1 and i B.(1 pt) Write an expression for h/ x at a point i+1/2 half way between i and i+1. C. (2 pt) Using your solutions from #2 and #3, write an expression for the second derivative of h with respect to x at point i. D. (2 pt) Substitute this expression into equation 1 and solve for h i . Consider a steady state 1-D flow problem representing a cross-section through an aquifer with a fully-penetrating lake or river on one side (head of 100 m) and a fully-penetrating lake or river on the other side (head of 200 m). Assume the aquifer is homogeneous. The length of the aquifer is 10 km, and it is discretized (divided) into 11 Excel cells ( x=1000 m). For this exercise, we’ll consider the “node” where the head to be determined is 1
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located at the center of each Excel cell. You know the head on the left and right hand side, and you have figured out (in #5) how to solve for head at any point. 3. (2 pts) Before you begin the numerical approximation: This problem has a simple analytical solution. What is the head at each cell? 100 ? ? ? ? ? ? ? ? ? 200 4. (2 pts) Type in your expression from #5 into all of the interior nodes (translating into an Excel formula). What happens and why? There are two methods to solve this in Excel. Both require “iterations”. The first method
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This note was uploaded on 11/29/2011 for the course GLY 6826 taught by Professor Screaton during the Fall '11 term at University of Florida.

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Assn1_2011 - GLY 6826 Assignment 1 Finite-difference...

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