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 Yun Chen 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..