1 The Collocation Method

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Frederick Harvey
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1 CS410 Assignment 7 Due: 1/5/14 (Fri) at 6pm You must wor eiter on your own or wit one partner. You may discuss bacground issues and general solution strategies wit oters, but te solutions you submit must be te wor of just you (and your partner). If you wor wit a partner, you and your partner must first register as a group in CMS and ten submit your wor as a group. Points may be deducted for poor style and recless inefficiency. Topics: Metods for Boundary Value Problems, B-spline review, matrix set-up 1 Te Collocation Metod In tis problem you are to compute an approximate solution ũ(x) to te boundary value problem u (x) + q(x)u(x) = r(x) a x b (1) u(a) = u a u(b) = u b () using te metod of collocation. Te idea is to express ũ(x) as a linear combination of simple basis functions. Te coefficients tat define te linear combination are ten determined via a linear system tat is obtained by imposing certain conditions on ũ(x). For basis functions we will use te B-splines B 0 (x),..., B n+1 (x) tat were introduced in A3. In particular, we will see an approximate solution to (1) of te form ũ(x) = n+1 α B (x) =0 were wit = (b a)/(n 1) and ( ) x x B (x) = B x = a + ( 1) = 0:n + 1. Recall tat A3 was about interpolation wit B-splines. In tat assignment we also determined α 0,..., α n+1 via a linear system solve. Te linear equations enforced interpolation conditions ũ(x i ) = y i i = 1:n. and a pair of end conditions, e.g., u (a) = 0, u (b) = 0. For te BVP (1)-() we proceed similarly. As in A3, te linear equations stipulate te properties tat we want ũ(x) to satisfy. Te first of tese is te left boundary condition: ũ(a) = ũ(x 1 ) = u a Next, we insist tat te differential equation is satisfied by ũ(x) at te collocation points x 1,..., x n : ũ (x i ) + q(x i )ũ(x i ) = r(x i ) i = 1:n. (3) Lastly, we want to be sure tat te rigt boundary condition is satisfied: You job is to implement a function tat does tis: ũ(b) = ũ(x n ) = u b 1

If you wor wit a partner, you and your partner must first register as a group in CMS and ten submit your wor as a group. Points may be deducted for poor style and recless inefficiency.

2 function alpa = BVP_Collocation(a,ua,b,ub,q,r,n) a<b and ua and ub are scalars. q and r are andles to functions q(x) and r(x) defined on [a,b]. n is a positive integer, n >=. alpa is a column (n+)-vector wit te property tat if utilde(x) = alpa(1)b_0(x) alpa(n+)*b_{n+1}(x) ten utilde(a) = ua -utilde (z(i)) + q(z(i))u(z(i)) = r(z(i)) i=1:n utilde(b) = ub were z = linspace(a,b,n) and B_{}(z) is te B-spline Bstar((z-x)/) were x = a+(-1) and = (b-a)/(n-1). (Sorry for te subscript-from-one annoyances.) A test script SowCollocation is provided tat can be used to compare your implementation wit an analogous procedure based on te metod of finite differences. (You will also need to download BVP FiniteDiff). In tis problem we are NOT concerned wit te efficient set up of te linear system tat specifies te α s. Assignment A3 gave you enoug practice wit tat, i.e., te exploitation of te local support feature of te B-spline basis. Clearly, tat tecnology can be exploited ere. Our goal is simply for you to appreciate te collocation framewor by using it to solve a simple problem. So tat you do not get bogged down in low-level details associated wit te evaluation of te B and teir derivatives, we supply te following function on te website: function [y,dy,ddy] = derbstar(z) z is a scalar y = Bstar(z) dy = Bstar (z) ddy = Bstar (z) (Note tat te equations in (3) involve second derivatives of te B evaluated at te x i.) Again, don t spend time vectorizing or exploiting te local support properties of te basis function just set up te linear system correctly and use \. Submit BVP Collocation to CMS. Te Cran-Nicolson Metod for te Heat Equation Here is a simple version of te eat equation: u(x, t) = u(x, t) + s(x, t) a x b, t 0 (4) t x Tin of u(x, t) as te temperature of a rod at time t were s(x, t) is a given eat-source function. Te temperature at te start is nown, u(x, t) = u (0) (x) a x b and remains te same at te endpoints u(a, t) = u (0) (a) = u a u(b, t) = u (0) (b) = u b t 0. (5) For us, te discretization of tis problem involves two parameters. One involves space and one involves time: = (b a)/(n 1) t > 0. Te goal is to produce approximations u(x, t j )

$.. + alpa(n+)*b_{n+1}(x) ten utilde(a) = ua -utilde (z(i)) + q(z(i))u(z(i)) = r(z(i)) i=1:n utilde(b) = ub were z = linspace(a,b,n) and B_{}(z) is te B-spline Bstar((z-x)/) were x = a+(-1) and =$

3 were x = a + ( 1) for = 1:n and t j = j t for j = 0, 1,,.... Since te value of u(x, t) is fixed at te endpoints (see equation (4)), we set 1 = u a n = u b j = 0, 1,,... Te Cran-Nicolson sceme relates te approximate solution at time t j+1 to te approximate solution at time t j as follows: = 1 t +1 u(j) u(j) u(j) u(j+1) 1 ( + s x, t j + ) t Assume tat we now, = 1:n and want to compute u(j+1), = 1:n. Of course, te endpoint values are nown, 1 = 1 = u a n = n = u b so it is all about computing,..., n 1 from nown stuff. Using te giant Cran-Nicolson divided difference recipe above, sow tat we ave an (n )-by-(n ) linear system of te form T... n 1 = rs tat involves u(j) 1,..., u(j) n,, t, s-evaluations Complete te following function so tat it carries out a Cran-Nicolson step function unext = CranN(uNow,a,b,n,tc,deltaT,s) unow is a column n-vector. a < b n is a positive integer tc is te current time. deltat >0 is te time step. s is a andle to a function of te form s(x,t) unext is a column n-vector wose entries satisfy unext(1) = unow(1), unext(n) = unow(n), and unew() - unow() 1 (unew(+1)-*unew()+unew(-1)) = deltat ^ 1 (unow(+1)-*unow()+unow(-1)) s(x(),tc+deltat/) ^ for =:n-1 were = (b-a)/(n-1); x = linspace(a,b,n); It is fine for you NOT to exploit T s sparse structure. Just set it up explicitly and use \. A test script SowHeat is provided. Submit CranN to CMS. 3

now, = 1:n and want to compute u(j+1), = 1:n. Of course, te endpoint values are nown, 1 = 1 = u a n = n = u b so it is all about computing,..., n 1 from nown stuff.

4 3 Te Sooting Metod for a Two-Point Boundary Value Problem Te function Cannon v0(v0) solves te A5 initial value problem ẋ = v(t) cos(θ(t)) ẏ = v(t) sin(θ(t)) θ = g/v(t) cos(θ(t)) v = D(t)/m g sin(θ(t)) were D(t) = cρs ((ẋ(t) + w) + ẏ ) and x(0) = 0, y(0) = 0, θ(0), and v(0) = v 0 are te given initial conditions. For a given input v0, te function displays a table tat reports just ow far te cannonball travels for various initial angles and constant wind speeds, e.g., v0 = Cannonball Distance as a function of initial angle A (degrees) and eadwind w A w = -0 w = -10 w = 0 w = 10 w = Develop an analogous function Cannon d(d) tat determines te required initial velocity v 0 so tat te cannonball travels exactly distance d before landing. Sample output: Required travel distance = Required initial velocity as a function of initial angle A (degrees) and eadwind w A w = -0 w = -10 w = 0 w = 10 w =

For a given input v0, te function displays a table tat reports just ow far te cannonball travels for various initial angles and constant wind speeds, e.g., v0 = 50.

5 To do tis you need to understand and mae use of te Matlab root-finder fzero. A simple example tells all. Suppose function z = MyF(x,a) z = a*x^ - 0; is available. We now tat if a = ten tis function as a single root in te interval [3,4]. Te following script assigns te root to r: a = ; L = 3; R = 4; Braceting_Interval = [L,R]; Contains te root of interest r = fzero(@(x) MyF(x,a),Braceting_Interval) Now ere is wat you do to compute te magic v 0, i.e., te initial velocity so tat wen te cannonball lands, x(t final ) = x final = d. Suppose F(v0,teta,w,d) is a function tat runs ode45 wit te same terminate-onlanding event function and wit initial condition [0;0;teta;v0]. If F returns te value of x final d, ten it evaluates to zero precisely wen v 0 is te required initial velocity, i.e., te initial velocity tat causes te terminating value of x to be d. Tus, to produce Cannon d(d) you need to adjust Cannon v0 so tat (a) it includes te subfunction F just described and (b) it replaces te function d = HowFar(teta,w,v0) wit a function v0 = HowFast(teta,w,d) tat returns te required initial velocity. HowFast is essentially a one-liner tat calls fzero. Tin a little bit about te required braceting interval tat you pass to fzero. Te smaller te interval te fewer te number of F-calls and tat means a reduced number of f-evaluations overall. Include comments on ow you pic te braceting interval. Note: fzero is unappy if tere is no root in te braceting interval. You may assume tat te incoming d satisfies 10 d 500. Submit Cannon d to CMS. 5

Te following script assigns te root to r: a = ; L = 3; R = 4; Braceting_Interval = [L,R]; Contains te root of interest r = fzero(@(x) MyF(x,a),Braceting_Interval) Now ere is wat you do to compute te

Verifying Numerical Convergence Rates

1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and