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

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.

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

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

How Approximate are these Solutions? Try an example:

Using The Forward Method

Using The Backward Method

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

Euler’s Forward Approximation Graphed (1)

Euler’s Forward Approximation Graphed (2)

Euler’s Forward Approximation Graphed (3)

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?

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

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

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

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

Exact Solution Graphed We see that

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.

Previous Discretized Equation

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

New Discretized Equation

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.

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.