g(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany

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1 Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required to mke solution of ODE unique In initil vlue problem ll side conditions specified t single point sy t Chpter Boundry Vlue Problems for Ordinry Differentil Equtions In boundry vlue problem (BVP) side conditions specified t more thn one point kth order ODE or equivlent first-order system requires k side conditions Copyright c 2 Reproduction permitted only for noncommercil eductionl use in conjunction with the book For ODE side conditions typiclly specified t two points endpoints of intervl [ b] so we hve two-point boundry vlue problem 2 Boundry Vlue Problems continued Generl first-order two-point BVP hs form y f(t y) <t<b with boundry conditions Exmple: Seprted Liner BC Two-point BVP for second-order sclr ODE g(y() y(b)) o where f: R n R n nd g: R 2n R n u f(t u u ) with boundry conditions <t<b Boundry conditions re seprted if ny given component of g involves solution vlues only t or t b but not both Boundry conditions re liner if of form B y()b b y(b)c where B B b R n n nd c R n u() α u(b) β is equivlent to first-order system of ODEs [ ] y y y 2 2 <t<b f(t y y 2 ) with seprted liner boundry conditions y () y (b) α y 2 () y 2 (b) β BVP is liner if both ODE nd boundry conditions re liner 3 4

2 Existence nd Uniqueness Unlike IVP with BVP we cnnot begin t initil point nd continue solution step by step to nerby points Insted solution determined everywhere simultneously so existence nd/or uniqueness my not hold For exmple u u with boundry conditions u() <t<b u(b)β with b integer multiple of π hs infinitely mny solutions if β but no solution if β Existence nd Uniqueness continued In generl solvbility of BVP y f(t y) with boundry conditions <t<b g(y() y(b)) o depends on solvbility of lgebric eqution g(x y(b; x)) o where y(t; x) denotes solution to ODE with initil condition y() xfor x R n Solvbility of ltter system is difficult to estblish if g is nonliner 5 6 Existence nd Uniqueness continued For liner BVP existence nd uniqueness re more trctble Consider liner BVP y A(t) y b(t) <t<b where A(t) nd b(t) re continuous with boundry conditions B y()b b y(b)c Let Y (t) denote mtrix whose ith column y i (t) clled ith mode is solution to y A(t)y with initil condition y() e i Then BVP hs unique solution if nd only if mtrix Q B Y ()B b Y(b) is nonsingulr 7 Existence nd Uniqueness continued Assuming Q is nonsingulr define Φ(t) Y(t)Q nd Green s function { Φ(t)B Φ()Φ G(t s) (s) s t Φ(t)B b Φ(b)Φ (s) t<s b Then solution to BVP given by y(t) Φ(t)c G(t s) b(s) ds This result lso gives bsolute condition number for BVP κ mx{ Φ G } 8

3 Conditioning nd Stbility Conditioning or stbility of BVP depends on interply between growth of solution modes nd boundry conditions For IVP instbility is ssocited with modes tht grow exponentilly s time increses For BVP solution is determined everywhere simultneously so there is no notion of direction of integrtion in intervl [ b] Growth of modes incresing with time is limited by boundry conditions t b nd growth of decying modes is limited by boundry conditions t Numericl Methods for BVPs For IVP initil dt supply ll informtion necessry to begin numericl solution method t initil point nd step forwrd from there For BVP we hve insufficient informtion to begin step-by-step numericl method so numericl methods for solving BVPs re more complicted thn those for solving IVPs We consider four types of numericl methods for two-point BVPs: Shooting Finite difference For BVP to be well-conditioned growing nd decying modes must be controlled ppropritely by boundry conditions imposed 9 Colloction Glerkin Shooting Method In sttement of two-point BVP we re given vlue of u() Shooting Method continued If we lso knew vlue of u () then we would hve IVP tht we could solve by methods previously discussed Lcking tht informtion we try sequence of incresingly ccurte guesses until we find vlue for u () such tht when we solve resulting IVP pproximte solution vlue t t b mtches desired boundry vlue u(b) β α β b For given γ vlue t b of solution u(b) toivp with initil conditions u() α u f(t u u ) u () γ cn be considered s function of γ syg(γ) Then BVP becomes problem of solving eqution g(γ) β One-dimensionl zero finder cn be used to solve this sclr eqution 2

4 Exmple: Shooting Method Consider two-point BVP for second-order ODE u 6t <t< u() u() For ech guess for u () we integrte ODE using clssicl fourth-order Runge-Kutt method to determine how close we come to hitting desired solution vlue t t We trnsform second-order ODE into system of two first-order ODEs y y2 y 2 6t We try initil slope of y 2 () Using step size h 5 we first step from t tot 5 Clssicl fourth-order Runge-Kutt method gives pproximte solution vlue t t y () y () h 6 (k k 2 k 3 k 4 ) ( ) Next we step from t 5tot 2 getting y (2) ( ) so we hve hit y () 2 insted of desired vlue y () We try gin this time with initil slope y 2 () obtining y () 5 ( ) y (2) ( 25 3 ) so we hve hit y () insted of desired vlue y () We now hve initil slope brcketed between nd We omit further itertions necessry to identify correct initil slope which turns out to be y 2 () : y () ( )

5 y (2) ( 75 3 ) so we hve indeed hit trget solution vlue y () st ttempt trget 2nd ttempt 5 Multiple Shooting Simple shooting method inherits stbility (or instbility) ssocited IVP which my be unstble even when BVP is stble Such ill-conditioning my mke it difficult to chieve convergence of itertive method for solving nonliner eqution Potentil remedy is multiple shooting in which intervl [ b] is divided into subintervls nd shooting is crried out on ech Requiring continuity t internl mesh points provides BC for individul subproblems Multiple shooting results in lrger system of nonliner equtions to solve 7 8 Finite Difference Method Finite difference method converts BVP into system of lgebric equtions by replcing ll derivtives by finite difference pproximtions For exmple to solve two-point BVP u f(t u u ) u() α <t<b u(b) β we introduce mesh points t i ih i n where h (b )/(n) We lredy hve y u() αnd y n u(b) β nd we seek pproximte solution vlue y i u(t i ) t ech mesh point t i i n Finite Difference Method continued We replce derivtives by finite difference quotients such s nd u (t i ) y i y i 2h u (t i ) y i 2y i y i h 2 yielding system of equtions ( y i 2y i y i h 2 f t i y i y ) i y i 2h to be solved for unknowns y i i n System of equtions my be liner or nonliner depending on whether f is liner or nonliner 9 2

6 Exmple: Finite Difference Method Consider two-point BVP Finite Difference Method continued u 6t <t< In this exmple system to be solved is tridigonl which sves on both work nd storge compred to generl system of equtions This is generlly true of finite difference methods: they yield sprse systems becuse ech eqution involves few vribles u() u() To keep computtion to minimum we compute pproximte solution t one mesh point in intervl [ ] t 5 Including boundry points we hve three mesh points t t 5 nd t 2 From BC we know tht y u(t ) nd y 2 u(t 2 ) nd we seek pproximte solution y u(t ) 2 22 Approximting second derivtive by stndrd finite difference quotient t t gives eqution ( y 2 2y y h 2 f t y y ) 2 y 2h Substituting boundry dt mesh size nd right hnd side for this exmple 2y (5) 2 6t or so tht 4 8y 6(5) 3 In prcticl problem much smller step size nd mny more mesh points would be required to chieve cceptble ccurcy We would therefore obtin system of equtions to solve for pproximte solution vlues t mesh points rther thn single eqution s in this exmple y(5) y /825 which grees with pproximte solution t t 5 tht we previously computed by shooting method 23 24

7 Colloction Method Colloction method pproximtes solution to BVP by finite liner combintion of bsis functions For two-point BVP u f(t u u ) <t<b u() α u(b) β we seek pproximte solution of form n u(t) v(t x) x i φ i (t) i where φ i re bsis functions defined on [ b] nd x is n-vector of prmeters to be determined Colloction Method Populr choices of bsis functions include polynomils B-splines nd trigonometric functions Bsis functions with globl support such s polynomils or trigonometric functions yield spectrl or pseudospectrl method Bsis functions with highly loclized support such s B-splines yield finite element method Exmple: Colloction Method Colloction Method continued Consider gin two-point BVP To determine vector of prmeters x define set of n colloction points t < < t n b t which pproximte solution v(t x) is forced to stisfy ODE nd boundry conditions u 6t <t< u() u() Common choices of colloction points include eqully-spced mesh or Chebyshev points Suitbly smooth bsis functions cn be differentited nlyticlly so tht pproximte solution nd its derivtives cn be substituted into ODE nd BC to obtin system of lgebric equtions for unknown prmeters x To keep computtion to minimum we use one interior colloction point t 5 Including boundry points we hve three colloction points t t 5 nd t 2 so we will be ble to determine three prmeters As bsis functions we use first three monomils so pproximte solution hs form v(t x) x x 2 tx 3 t

8 Derivtives of pproximte solution function with respect to t re given by v (t x) x 2 x 3 t v (t x) 2x 3 Requiring ODE to be stisfied t interior colloction point t 2 5 gives eqution or v (t 2 x) f(t 2 v(t 2 x)v (t 2 x)) 2x 3 6t 2 6(5) 3 Left boundry condition t t gives eqution x x 2 t x 3 t 2 x nd right boundry condition t t 3 gives eqution x x 2 t 3 x 3 t 2 3 x x 2 x 3 29 Solving this system of three equtions in three unknowns gives x x 2 5 x 3 5 so pproximte solution function is qudrtic polynomil u(t) v(t x) 5t5t 2 At interior colloction point t 2 5 we hve pproximte solution vlue u(5) v(5 x) 25 which grees with solution vlue t t 5 obtined previously by other two methods Glerkin Method Rther thn forcing residul to be zero t finite number of points s in colloction we could insted minimize residul over entire intervl of integrtion For exmple for sclr Poisson eqution in one dimension u f(t) <t<b with homogeneous boundry conditions u() u(b) subsitute pproximte solution n u(t) v(t x) x i φ i (t) i into ODE nd define residul r(t x) v n (t x) f(t) x i φ i (t) f(t) i 3 Glerkin Method continued Using lest squres method we cn minimize F (x) 2 r(t x)2 dt by setting ech component of its grdient to zero which yields symmetric system of liner lgebric equtions Ax b where ij φ j (t)φ i (t) dt nd b i f(t)φ i (t) dt whose solution gives vector of prmeters x More generlly weighted residul method forces residul to be orthogonl to ech of set of weight functions or test functions w i ie r(t x)w i(t) dt i n which yields liner system Ax b where now ij φ j (t)w i(t) dt nd b i i(t) dt whose solution gives vector of prmeters x 32

9 Glerkin Method continued Glerkin Method continued Mtrix resulting from weighted residul method is generlly not symmetric nd its entries involve second derivtives of bsis functions Both drwbcks overcome by Glerkin method in which weight functions re chosen to be sme s bsis functions ie w i φ i i n Orthogonlity condition then becomes r(t x)φ i(t) dt i n or v (t x)φ i (t) dt i(t) dt i n 33 Degree of differentibility cn be reduced using integrtion by prts which gives v (t x)φ i (t) dt v (t)φ i (t) b v (t)φ i (t) dt v (b)φ i (b) v ()φ i () v (t)φ i (t) dt Assuming bsis functions φ i stisfy homogeneous boundry conditions so φ i () φ i () orthogonlity condition then becomes v (t)φ i (t) dt i(t) dt i n which yields system of liner equtions Ax b with ij φ j (t)φ i (t) dt nd b i i(t) dt whose solution gives vector of prmeters x A is symmetric nd involves only first derivtives of bsis functions 34 Exmple: Glerkin Method Consider gin two-point BVP u 6t <t< u() u() Thus we seek pproximte solution of form u(t) v(t x) x φ (t)x 2 φ 2 (t)x 3 φ 3 (t) We will pproximte solution by piecewise liner polynomil for which B-splines of degree ( ht functions) form suitble set of bsis functions To keep computtion to minimum we gin use sme three mesh points but now they become knots in piecewise liner polynomil pproximtion φ 5 φ 2 5 φ From BC we must hve x nd x 3 To determine remining prmeter x 2 we impose Glerkin orthogonlity condition on interior bsis function φ 2 nd obtin eqution ( 3 ) j φ j (t)φ 2 (t) dt x j 6tφ 2(t) dt or upon evluting these simple integrls nlyticlly 2x 4x 2 x 3 3/2 36

10 Substituting known vlues for x nd x 3 then gives x 2 /8 for remining unknown prmeter so piecewise liner pproximte solution is u(t) v(t x) 25φ 2 (t)φ 3 (t) 5 5 We note tht v(5 x) 25 which gin is exct for this prticulr problem More relistic problem would hve mny more interior mesh points nd bsis functions nd correspondingly mny prmeters to be determined Resulting system of equtions would be much lrger but still sprse nd therefore reltively esy to solve provided locl bsis functions such s ht functions re used Resulting pproximte solution function is less smooth thn true solution but becomes more ccurte s more mesh points re used Eigenvlue Problems Stndrd eigenvlue problem for second-order ODE hs form u λf(t u u ) u() α <t<b u(b) β where we seek not only solution u but lso prmeter λ s well Sclr λ (possibly complex) is eigenvlue nd solution u corresponding eigenfunction for this two-point BVP Discretiztion of eigenvlue problem for ODE results in lgebric eigenvlue problem whose solution pproximtes tht of originl problem Exmple: Eigenvlue Problem Consider liner two-point BVP u λg(t)u < t < b u() u(b) Introduce discrete mesh points t i in intervl [ b] with mesh spcing h nd use stndrd finite difference pproximtion for second derivtive to obtin lgebric system y i 2y i y i h 2 λg i y i i n where y i u(t i ) nd g i g(t i ) nd from BC y u() nd y n u(b) 39 4

11 Assuming g i divide eqution i by g i for i n to obtin liner system Ay λy where n n mtrix A hs tridigonl form 2/g /g A /g 2 2/g 2 /g 2 h 2 /g n 2/g n /g n /g n 2/g n This stndrd lgebric eigenvlue problem cn be solved by methods discussed previously 4