5 Ordinary Differential Equations: Finite Difference Methods for Boundary Problems

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1 5 Ordinary Differential Equations: Finite Difference Metods for Boundary Problems Read sections 10.1, 10.2, 10.4 Review questions , , Introduction In te previous capters we investigated initial value problems: A process or a state was described by a system of ordinary differential equations y f t y t t 0 T y t m Te general solution of suc a problems contains m constants of integration wic were defined by imposing additional conditions at te point t 0, y i t 0 y 0 i i 1 m or, in vector notation, y t 0 For example, consider te scalar second-order equation u t u u If we transform tis equation to te equivalent first-order system, we obtain a system of two equations, u y u f y t y 2 t y 1 y 2 Te initial values become u t 0 y 0 u 0 u t 0 Instead of determining te two constants of integration in tis way one could equally well require two different conditions, for instance, u t 0 α u T Since tere is no qualitative difference between te two boundary points any longer, we will use a more suggestive notation: a t 0 and b T. A problem of te type u t u u t u a α u b 1 ū 0 β a b β

2 is called a (two point) boundary value problem, because te additional conditions for determining te constants of integration are given at te boundary of te domain of definition Ω a b. We know already tat first-order systems of differential equations include iger order equations as a special case. Terefore, te most general form of a boundary value problem is given by y f t y t g y a y b a b 0 Since boundary value problems very often appear in te form of second-order equations, we will investigate only suc systems in te following Two Examples A Stationary Heat Equation We investigate te temperature distribution in a long and tin rod wit lengt L and a constant cross section. Assume tat te eat transfer properties are independent of te position. Distributed over te rod is a eat source. In te equilibrium state, te differential equation ku q x x 0 L olds true, were u x describes te temperature of te rod at point x. If we fix te temperature at te two endpoints at T 0 and T L, respectively, we obtain te boundary conditions u 0 T 0 u L Deformation of a Beam A beam wit lengt L rests at its two ends in fixed positions. A distributed force is applied to te bar. Te deformation y x of te bar from its rest position can be describes by a system of two second-order differential equations, d 2 M dx 2 f x d 2 y M x dx 2 EI were M x is te bending moment, and E and I are constants (te elasticity module and te surface moment of inertia, respectively). Te boundary conditions are M 0 y Existence and Uniqueness of Solutions M L y L Tere is a fundamental difference between initial and boundary value problems wic is important for bot te teoretical investigation and te numerical approximation. Assume tat te 0 0 T L 2

3 rigt-and side f of te differential equation is smoot enoug (for example continuously differentiable 1 ), ten te solution of te initial value problem always exists and is unique at least over sufficient small time intervals. Suc a property does no longer old for boundary value problems. Te following two examples illustrate some possibilities. Example 5.1. Consider te boundary value problem subject to te boundary conditions u u t u 0 0 u b Te general solution of te ordinary differential equation satisfying u 0 0 is u t csint for any constant c. If b is an integer multiple of π, ten csinb 0 for any c, so tere are infinitely many solutions of te boundary value problem if β 0, but tere is no solution if β 0. Example 5.2. Te problem as two solutions of te form were θ satisfies u t u e u 1 0 t u 0 u 1 0 b β ln cos t 1 2 θ 2 cos θ 4 θ 2ecos θ 4 Tis nonlinear equation as exactly two solutions for θ. Te corresponding solutions are plotted in Figure Finite Difference Metods for Linear Problems Discretization A second-order ordinary differential equation u t u u is called linear if te equation as te form u b t u c t u d t Nonlinear problems will be investigated in later capters. In te beginning we assume tat b t 0 t a b 1 Tis requirement is a severe restriction in practice. Fortunately, it can be considerably relaxed. 3

4 Figure 1: Te two solutions of te problem in Example 5.2 Te boundary conditions are u 0 α u b β We coose a step size and subdivide te interval a b into n 1 subintervals I i t i t i 1 for i 0 n suc tat t i a i i 0 n 1 b a n 1 Te points t i are called grid points or nodes. t 0 a and t n 1 b are te boundary nodes. Exactly as before we try to approximate te exact solution u t i by values u i wic are derived by finite difference approximations. In order to obtain an approximation of te second derivative we start by approximating te first derivative: Te next step is Using te abbreviations c i u t i u t i u t i c t i and d i 1 2 u t i 1 2u t i 2 u t i u t i 1 2 u t i 1 u t i u t i 2 u t i 2 u t i 1 2u t i u t i 1 2 d t i we obtain u t i 1 c i u t i 4 d i i 1 n

5 Te numerical approximation is obtained by replacing te approximation sign by an equality sign: u i 1 2u i u i 1 c i 2 u i 2 d i i 1 n Note tat we multiplied troug te equation by 2. It is more tan only convention tat te minus sign is used in tis transformation. Te remaining unknowns are simply obtained by applying te boundary conditions, u 0 α u n 1 β Finally, we obtain a linear system of equations wic can be conveniently written down in matrix notation: 2 c c A c n 2 u 1 u 2.. u n u If all of te c i s are equal to zero, te resulting matrix becomes 2 d 1 2 d 2. 2 d n 1 2 d n f α β Tis is a matrix wic we will observe in many different applications. So it is wort remembering it carefully! Te system matrix is a tridiagonal matrix. Tis is a very special form of sparse matrices. Te latter notion denotes matrices wic consist almost exclusively of zero entries. It is obvious tat one sould take care of suc a property because tis will save a lot of computation time and memory requirements. Remark 5.1. MATLAB provides many convenient functions for andling sparse matrices. Once tey are instantiated, all standard operators take care of tis special property. I recommend to ave a look at te MATLABexercises sparse matrices (no. 4) of period 1 once again. Te matrix A as also oter very interesting properties: Te matrix is symmetric. If c i 0 for all i 1 n, te matrix is positive definite. If we would not ave multiplied by minus 1 wen deriving te linear system, te matrix would be negative definite (wic is someow inconvenient). 5

6 Te first property is easy to see, wile te proof of te second one requires a little bit of computations. Tere is a small detail wic makes a considerable difference compared to finite difference metods in initial value problems: In initial value problems, te approximations u i could be determined one after te oter in a sequential process. Tis is no longer possible for boundary value problems. One must solve te linear system of equations and obtains te solutions u i simultaneously Discretization Errors and Accuracy As usual we are interested in estimating te global error e i u i u t i and its maximal value e max e i 1 i n Tis is a ard problem. Opposed to tat, it is relatively easy to derive an estimate for te local error L i wic appears if we insert te exact solution into te discrete equations: L i 1 2 u t i 1 2u t i If we use te differential equation u t i L i u t i 1 c t i u t i d t i i 1 n c t i u t i 1 2 u t i 1 2u t i As usual, let us apply te Taylor expansions u t i 1 u t i u t i 2 2 u t i d t i, we obtain te expression u t i 1 u t i 3 6 u t i 4 24 u 4 s i 1 Tis yields 1 L i 12 u 4 τ i 2 τ i t i 1 t i 1 Te problem consists now of deriving a relation between te local and te global error. Tis can be done by subtracting te two equations from eac oter: 1 2 u i 1 2u i 1 2 u t i 1 2u t i u t i 1 c i u t i 1 2 e i 1 2e i e i 1 c i e i 6 u i 1 c i u i L i i d i d i L i 1 n

7 Here, we ave used te definition e i u i u t i. Multiplication by 2 yields e i 1 2 c i 2 e i e i 1 2 L i i 1 n Because te values for u 0 and u n 1 are exact, we ave e 0 e n 1 0. Summarizing we obtain te linear system of equations A e 2 L If te matrix A is nonsingular, it olds Tis gives rise to te estimate e e 2 A 1 2 A 1 L L In order to prove te convergence of te metod one needs to know ow A 1 depends on. Tis estimation is nontrivial. Note tat te dimension of A 1 depends on! Teorem 5.1. If c t 0 for all t a b, ten and, consequently, A 1 e O 2 O 2 Tis means tat te order of discretization and te order of convergence are identical. Note tat te teorem contains a typical stability result: If te perturbations (ere: L ) are small, ten te error of te result (ere: e) remains small. For later considerations, it sould be noted tat κ A A 1 Tis property will be essential in later sections Te Discretization of Convection Terms O 2 Te first-order term b t u in a second-order differential equation u b t u c t u d t is often called te convection term because it very often models pysical convection in a system. We already know finite difference approximation to first-order derivatives, namely, u t i D u t i : u t i u t i 1 7

8 and u t i D u t i : u t i 1 u t i respectively. Tey are called backward and forward finite differences. Moreover, we know tat bot approximations ave only first order of accuracy, D u t i u t i O 1 D u t i u t i O 1 Since we already ave second-order approximations for all oter terms, it would be wise to use a second-order approximation ere, too. Tis can be acieved if we use, for example, te mean between forward and backward differences: D 0 u 1 t i D u t i 2 D u t i u t i 1 u t i 1 2 It is easy to see tat tis central difference as second order (exercise!), D 0 u t i u t i O 2 In a similar manner as above, we obtain te difference equation 1 b i 2 u i 1 2 c i 2 u i Let us introduce te abbreviations p i 2 c i 2 q i Tis gives rise to te linear system of equations p 1 q 2 r 1 p qn A r n 1 p n 1 b i 2 u i 1 1 b i 1 r i 2 u 1 u 2.. u n u d i i b i 1 2 d 1 d n r 0 α 2 d 2. d n 1 q n 1β 2 f 1 n It is very ard to sow te convergence for suc a system. However, one can sow tat A is nonsingular and e O 2 if te continuous problem is uniquely solvable and te step size is sufficiently small. Te latter can really be a serious restriction! 8

9 5.2.4 Oter Boundary Conditions Wen deriving te discrete problem A u ad te form u a f we always assumed tat te boundary conditions α u b Many problems include boundary conditions wic contain derivatives of te solution, for example, u a α Tis is te case in te eat equation if we ave perfectly isolating boundaries or boundaries wic allow for eat transfer. Te discretization of te differential equation for i 1 is r 0 u 0 p 1 u 1 q 2 u 2 2 d 1 Wit te former kind of boundary conditions it was easy to replace te value for u 0 by α. Here, tis is no longer possible since u a must be discretized additionally. One metod can be derived as follows. We write down formally te discretization for i 0, r 1u 1 p 0 u 0 q 1 u 1 2 d 0 and a central discretization for te boundary condition, u 1 u 1 2 u 1 is a fictious value because it does not correspond to any value of te exact solution. But we can eliminate tis value, u 1 u 1 2α suc tat te difference equation, for i 0, becomes 2α p 0 u 0 q 1 u 1 2 d 0 or, equivalently, r 1 u 1 p 0 u 0 r 1 q 1 u 1 α β 2 d 0 2r 1α Tis equation is added to te old system suc tat u 0 can be computed. Te same idea can obviously be applied if te boundary condition at t b reads u b β. In te general case, tis discretization as only first order. But one can sow tat second order is obtained if α 0. Tere are even more general boundary conditions wic contain combinations of function values and values of te derivatives. Te approximation principles of tese conditions are exactly as before. Te following notions are often used to caracterize te different types of boundary conditions: 9

10 notion type example Diriclet fix a function value u a α Neumann fix a derivative value u a α Robin (mixed) linear combination of η 1 u a η 2 u a function values and derivatives α Higer Order metods Tere also exist iger order metods. Tey can be constructed in almost te same way as for finite difference metods using te Runge-Kutta idea or te multistep idea.matlab s solver for two-point boundary value problems is bvp4c. It implements a fourt-order Runge-Kutta metod wit automatic step size control. For second-order scalar boundary value problems it is more common to use multistep discretizations. 5.3 Examples A Simple Problem Consider te problem u 12t 2 t 0 1 u 0 0 u 1 1 Te exact solution is obviously u t t 4. We apply te finite difference metod wit different step sizes and compute te accuracy of te resulting approximation. Te MATLABcode is given on te next page. Te following table contains te results. q denotes te error reduction factor between two successive rows. According to our teory, it sould approac 4 since te metod is of second order. Te results confirm te teory. e q 1/ / / / / / / /

11 clear % Define number of cycles L = 8; l = zeros(l,1); el = zeros(l,1); % Cycle over all gridsizes n = 1; for l = 1:L % Define grid a = 0; b = 1; = (b-a)/(n+1); tv = linspace(a,b,n+2) ; ti = tv(2:end-1); % Define matrix diagonals p = 2*ones(n,1); q = -ones(n,1); r = -ones(n,1); % Define matrix A (Att.: Sparse matrix) A = spdiags([r,p,q],[-1,0,1],n,n); % Rigt-and side vector f = (-ˆ2)*12*ti.ˆ2; % Modify for boundary conditions % f(1) = f(1)+0; f(end) = f(end)+1; % Solve for approximate solution u = A f; % Compute te exact solution uex = ti.ˆ4; % Te error e = norm(u-uex,inf); % Build table l(l) = ; el(l) = e; % Prepare next step n = 2*n+1; end % Results l el q = el(1:end-1)./el(2:end) 11

12 5.3.2 A More Complex Example Te problem is given by te equation 1 t 2 u 2 6t 2 2t cost 1 t 2 sint t 0 1 subject to te boundary conditions u 0 1 u 1 2 sin1 Te rigt-and side and te boundary conditions were cosen in suc a way tat te (unique) exact solution becomes u t t 2 sint 1 Te first step consists of a reformulation of te problem in standard form. By differentiation we obtain wit u 2t 1 t 2 u d t d t 2 6t 2 2t cost 1 t 2 sint Te linear system can be constructed and solved. A compact form of te algoritm can be found on te following page. Te resulting table is given below: e q 1/ / / / / / /

13 clear L = 7; l = zeros(l,1); el = zeros(l,1); n = 3; for l = 1:L a = 0; b = 1; = (b-a)/(n+1); tv = linspace(a,b,n+2) ; ti = tv(2:end-1); p = 2*ones(n,1); % c = 0 b = (/2)*((-2*tv)./(1+tv.ˆ2)); q = -ones(n+2,1)+b; r = -ones(n+2,1)-b; A = spdiags([r(3:end),p,q(1:end-2)],[-1,0,1],n,n); f = (-ˆ2)*((2+6*ti.ˆ2+2*ti.*cos(ti))./(1+ti.ˆ2)-sin(ti)); f(1) = f(1)-r(2)*1; f(end) = f(end)-q(end-1)*(2+sin(1)); u = A f; uex = ti.ˆ2+sin(ti)+1; e = norm(u-uex,inf); l(l) = ; el(l) = e; n = 2*n+1; end l el q = el(1:end-1)./el(2:end) In order to compare tese results wit tose from MATLAB s built-in function bvp4c, we formulated te problem as a first-order system and required a tolerance of 10 7 wic is comparable to te last row of te previous table. Tis code needs only 28 grid points (compared to 256 in our previous attempt) to obtain an accuracy of ! For completeness, te programs are included: 13

14 clear solinit.x = linspace(0,1,11); solinit.y = zeros(2,lengt(solinit.x)); options = bvpset( RelTol,1e-7, AbsTol,1e-7,... BCJacobian,[1,0;0,0],[0,0;1,0], Stats, on ); sol = bvp4c(@funkomp,@bvkomp,solinit,options); tex = sol.x; uex = tex.ˆ2+sin(tex)+1; e = norm(sol.y(1,:)-uex,inf) % % Tis function to be placed in a separate file! function yp = funkomp(t,y) yp = zeros(2,1); yp(2) = -2*t*y(2)/(1+tˆ2)+(2+6*tˆ2+2*t*cos(t))/(1+tˆ2)-sin(t); yp(1) = y(2); % % Tis function to be placed in a separate file! function bv = bvkomp(ya,yb) bv = zeros(2,1); bv(2) = yb(1)-2-sin(1); bv(1) = ya(1)-1; 14