Numerical Methods 1. February 20, Joel Franklin, Cambridge University Press, 2013 and is used with the author s permission.

Transcription

1 Numerical Methods 1 February 20, Some of this material appears in Computational Methods for Physics by Joel Franklin, Cambridge University Press, 2013 and is used with the author s permission.

2 1.1 Root-Finding Given a function F (x), find some or all of the set { x i } n i=1 such that F ( x i)= 0. This defines the root-finding problem this type of uestion shows up in a variety of settings E&M example For the time-dependent fields in electricity and magnetism, we know that information (in that setting, the magnitude and direction of E and B at a particular point r = x ˆx + y ŷ + z ẑ at time t) travels at a finite and specific speed: c (in vacuum). The immediate implication is that if we would like to know the electric field at our observation point (and time) due to a charge that is moving along some prescribed trajectory: w(t), we need to evaluate the location of the particle not at time t, but at some earlier time t r (the retarded time). The setup is shown in Figure 1.1 the particle of charge is at w(t) at time t, but it is the earlier location w(t r ) that has information traveling at c to the observation point r at time t. The time it takes for the information from w(t r ) to reach r is t t r, and during that interval, the field information travels at c, so the distance travelled is c (t t r ). At the same time, the distance between r and w(t r ) is, geometrically, r w(t r ) puttingthese together, we have the defining euation for the retarded time: c (t t r )= r w(t r ) p (r w(t r )) (r w(t r )). (1.1) Notice that the retarded time t r is a function of t and r (the observation time and location), and is defined implicitly in terms of w(t r ). We may or may not be able to solve this euation for t r given w analytically (cannot be done for all but the simplest trajectories), but we can define a root-finding problem: F (T )=c (t T ) r w(t ). (1.2) The roots of F (T ) will provide the retarded time Method The procedure we will use is called bisection start with a pair of values, x 0` and x0 r with x 0` <x0 r (hence the ` and r designations) and F (x 0`) F (x0 r) < 0 1

3 ẑ w(t) w(t r ) ˆx r at t ŷ Figure 1.1: A particle with charge moves along the prescribed curve w(t). For the point r at time t, field information comes from the earlier time t r, when the charge was at w(t r ). (so that a root is in between the points). Then calculate the midpoint between this pair, x 0 m 1 2 x0` + x0 r, and evaluate the product p F (x 0`) F (x0 m). If p is less than zero, then the root lies between the left and middle points, so we can move the left and right points over by setting x 1` = x0`, x1 r = x 0 m.if p is greater than zero, the root is between the middle and right points, and we update x 1` = x0 m, x 1 r = x 0 r. The iteration is continued until the absolute value of F at the current midpoint is smaller than some tolerance, F (x n m) apple, (1.3) and then x n m is an approximation to the root. The process is shown pictorially in Figure 1.2, and is described with pseudocode in Algorithm Example the electric potential of a moving charge Given a vector describing the location of a charge at time t, w(t), we can use root-finding to find the retarded time associated with a field point r at time t. From that, we can evaluate the full electric field, or, because it is a 2

4 F (x) x x 0 x 0 m x 0 r x 1 x 1 m x 1 r Figure 1.2: Two iterations of the bisection procedure. Algorithm 1.1 Bisection(F, x 0`,x0 r, ) x` x r x 0` x 0 r x m 1/2(x` + x r ) while F (x m ) > do if F (x m ) F (x r ) < 0 then x` else x r x m x m end if x m 1/2(x` + x r ) end while return x m 3

5 scalar, we can focus on the potential: V (r,t)= r w(t r ) (r w(t r )) ẇ(t r )/c, (1.4) which reduces, for slow source-charge motion, to V (r,t)= 4 0 r w(t r ) (1.5) (we ll use this approximation). Given a charge s trajectory through space, w(t), we can use the routine described in Algorithm 1.2 to generate the (approximate) electric potential at an individual field point, and by evaluating the potential at a variety of points (in a grid, for example), we can generate a contour plot of it (that process is described in Problem 1.3 below). Algorithm 1.2 Vfield(x, y, z, t, w, ) r {x, y, z} F (X) c (t X) r w(x) tr p Bisection(F, 1,t, ) R (r w(tr )) (r w(t r )) 4 0 R V return V Problem 1.0 When you make a function in Mathematica (using Module), you cannot assign values to an argument of the function. To see what happens (and gain familiarity with the arcane error messages Mathematica has in store for you), write a function that adds two numbers and assigns the result to one of the inputs as follows: addem[a, b ] := Module[{}, a = a + b; Return[a];]. Try it out: addem[5,3] whathappens? Another potential pitfall in writing functions can occur when you try to return a variable that has no assigned value. Let s rewrite the addem function so that the return value is not assigned: addem[a, b ] := Module[{retval, c}, c = a + b; Return[retval];]. What output does addem[5,3] produce this time? 4

6 Problem 1.1 Implement the bisection procedure, and test it by finding the roots of f(x) = (x )(x+1) and the first two roots of J 0 (x) (obtainable via BesselJ[0,x] ) what are those roots? Use = as the tolerance in all cases. Problem 1.2 A charged particle moves along the trajectory described by w(t) =R cos(!t) ˆx + d!tẑ. (1.6) Use the bisection routine to find the retarded time for the point r =0at t =0 if R = 10 5 m,! = /s, d =1m. Use = as your tolerance. What is w(t r )? How does this compare with w(0), the value we would use if we ignored the retarded time? Problem 1.3 Given w(t) =d cos(!t) ŷ with d =.05 m,! =5s 1,andtaking/(4 0 )=1, c =1for simplicity, implement Algorithm 1.2 to find V and make contour plots of its value for points between 1! 1 in x and.1! 1 in y, withz =0and t going from 0 to 8 /5 in steps of 2 /(25) using: data=table[contourplot[vfield[x,y,0,t,w,eps],{x,-1,1},{y,.1,1}], {t,0,8 Pi/5, (2 Pi/5)/5}] You can animate these as a flip book via ListAnimate[data]. Use = again, and take 1 = 10 for the bisection inside Algorithm