1. Finding A Better Final Discretized Equation
Bob Reasey
March 25, 2002

2. Goals
Observe why the concentration at time t in the discretized equation is not accurate for all choices of t.
Find a better approximation for the concentration at time t.

3. Methods of Approximation
Euler’s Forward Method
Euler’s Backwards Method

4. Major Differences Euler’s Forward Method Explicit Method
Uses the approximation at the pervious time interval
Euler’s Backward Method
Implicit Method
Uses the approximation at the current time interval
At each step, at least one linear equation must be solved at each step

5. How Approximate are these Solutions?
Try an example:

6. Using The Forward Method

7. Using The Backward Method

8. Actual Solution
By solving the equation by means of separable equations, the actual solution is found:

9. Euler’s Forward Approximation Graphed (1)

10. Euler’s Forward Approximation Graphed (2)

11. Euler’s Forward Approximation Graphed (3)

12. Problem
The different values of h yield different limits as h approaches infinity.
Based upon these three selections for h, what is the real limit?

13. Euler’s Backward Approximation Graphed
Red : h = .5
Blue : h = 1.5
Green : h = 2.5

14. Make Note
Notice that for all selected values of h, all approximations of approach 1 as n approaches infinity.

15. Summary of Tested Values
Forward
h=.5
h=1.5
h=2.5
Backward
h=.5
h=1.5
h=2.5

16. What happens as ? Forward If 0<h<2 If h<=0 or 2<=h
Backward
FOR ALL h!!

17. Exact Solution Graphed
We see that

18. Observations
Euler’s forward method is only accurate for certain values of h.
Euler’s backward method is accurate for all values of h; therefore, this is the more accurate and better approximation.

19. Previous Discretized Equation

20. Finding A Better Approximation
The previous equation is found through Euler’s forward method.
We need to change the equation so it follows Euler’s backwards method.
-This is done by substituting t = t + 1

21. New Discretized Equation

22. Is The New System Balanced?
As of now, the new System is not balanced.
LHS accounts for only interior nodes.
RHS accounts for nodes interior and on the boundary.
This system is true for each interior node.

23. Balancing The System
To balance the system, the LHS needs to account for boundary nodes.
Boundary conditions: c(x, y, t) = 0 for x, y on our boundary at any time, t.