FINITE DIFFERENCE METHOD

Alannah Bennie, Yun Chen, David Mantell, Fung Yee Ma OUTLINE

 What are Finite Difference Methods?  Background  Taylor Series Expansion of a Polynomial  First derivative of a function  Second derivative of a function  What is the Heat Equation?  Explicit Method for solving Heat Equation FINITE DIFFERENCE METHOD

Finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.

Example: the forward difference equation for the first derivative, as we will see, is: TAYLOR’S THEOREM

• Named after Brook Taylor, 1712

• gives approximations of a differentiable function around a given point.

• gives accurate estimations on the size of error in approximation TAYLOR’S THEOREM

If n ≥ 0 is an integer and ƒ is a function which is n times continuously differentiable on the closed interval [a, x] and n + 1 times differentiable on the open interval (a, x), then

where Rn(x) is the remainder and (n) f (xn) is the nth derivative of f evaluated at xn DERIVATION FROM TAYLOR’S THEOREM

 The first derivative of function f at x0 is:  f (x + h) = f (x ) + f ’(x )h + R (x ) 0 0 0 1 0 ChenYun  The derivative at a is:

 f ’(a + h) = f (a) + f ’ (a)h + R1(a)

 f ’ (a + h) – f (a) –R1(a) = f ’(a)h  Divide h on both sides, we get:

 When R1(a) is sufficiently small, FINDING THE SECOND DERIVATIVE

We have seen that the first derivative can be given by:

Similarly, we can do the same to find the second derivative. SECOND DERIVATIVE

Using the Taylor Series to approximate the derivative of a function, we have:

We can assume that R is small enough that we can ignore it. So with slight manipulation we have: HEAT CONDUCTION EQUATION

The Heat Conduction Equation: 2 α Uxx=Ut

is an important partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time. SOLVING THE HEAT EQUATION

 One way to numerically solve this equation is to approximate all the derivatives by finite differences.

 We partition the domain in space using x0,...,xJ and in time using t0,....,tN.

 We let the difference between two consecutive space points be h and between two consecutive time points be k. EXPLICIT METHOD

Using a forward difference we can get the recurrence equation:

Where, and

Notation: represents the numerical approximation at space j and time n+1. EXPLICIT METHOD

So, knowing the values at time n you can obtain the corresponding ones at time n+1 using this recurrence relation.

(r = k / h2) Notice: This equation states that the temperature at a location and time step is a linear combination of the prior temperatures at the location and at the nearest neighbors. EXPLICIT METHOD

Hence, if we know values for u at t = 0, then we could use the recurrence relation to calculate

values at t = t1. Then by repeating the process, we can get values for each successive time step.