Numerical Analysis II. Problem Sheet 8

Transcription

1 P. Grohs S. Hosseini Ž. Kereta Spring Term 15 Numerical Analysis II ETH Zürich D-MATH Problem Sheet 8 Problem 8.1 The dierential equation Autonomisation and Strang-Splitting ẏ = A(t)y, y(t ) = y, A(t) = ta, A = can be autonomised by introducing a variable z := ż = F (z) = }{{} =:(z) [ ] 1 1 [ ] y, which yields the ODE t (8.1.1) [ ] A(t)y. (8.1.) 1 (8.1a) Formulate the discrete evolution o a Strang splitting method or (8.1.) corresponding to the decomposition [ ] [ ] A(t)y F (z) = + 1 }{{} =:g(z) based on the exact evolutions or and g. HINT: Use the representation o the exact solution o an IVP or a homogeneous linear dierential equation. Solution: The exact evolutions to the linear homogeneous dierential equations ż = (z) and ż = g(z) are given by [ ] [ ] [ ] [ ] y y y e Φ h = and Φ h g = ha(t) y. t t + h t t The discrete evolutions o the corresponding Strang splitting method is now given by Ψ h Fz = Φ h/ Φ h g Φ h/ z. We thus obtain [ ] [ ] [ ] y Ψ h F = Φ h/ t Φ h g Φ h/ y = Φ h/ y t Φ h g t + 1h [ ] [ ] = Φ h/ e ha(t+ 1 h) y e ha(t+ 1 t + 1h = h) y. t + h Problem Sheet 8 Page 1 Problem 8.1

2 (8.1b) Implement the resulting method by completing the MATLAB template [t, y] = strangaut(y, t, T, h) HINT: The matrix exponential unction can be computed using expm. Solution: See strangaut.m, Listing 8.1. Listing 8.1: Implementation o strangaut.m. 1 u n c t i o n [t, y] = strangaut(y, t, T, h) % Strang splitting method or the solution o an autonomized IVP 3 % 4 % Input: 5 % y: initial value 6 % t: initial time 7 % T: end time 8 % h: step size 9 % 1 % output: 11 % t: time grid 1 % y: solution at the points o the grid % initialization 15 A = [, 1; -1, ]; % adapt step size, so that end time is attained precisely 18 N = c e i l ((T-t) / h); 19 h = (T-t) / N; 1 % allocate memory or results t = z e r o s (N+1, 1); 3 y = z e r o s (, N+1); 4 5 % set starting values 6 t(1) = t; 7 y(:, 1) = y; 8 9 % Timestepping 3 o r k = 1:N 31 3 % apply Strang splitting method y = expm(h * (t +.5*h) * A) * y; 35 t = t + h; 36 % save results 37 t(k+1) = t; 38 y(:, k+1) = y; Problem Sheet 8 Page Problem 8.1

3 39 end (8.1c) Plot the numerical solution obtained by applying the splitting method to the IVP (8.1.1) where y = [1, ], t =, T = 1, h =.1. To do so, complete the template strangautplot.m. Use time t as your z-coordinate. HINT: You might ind the unction plot3 to be o use. Solution: See strangautplot.m, Listing 8.. The resulting plot is shown in Figure 8.1. Listing 8.: Implementation o strangautplot.m. 1 u n c t i o n strangautplot % plot results rom strangaut 3 4 % initialization 5 t = ; 6 T = 1; 7 h =.1; 8 y = [1; ]; 9 1 % compute numerical solution 11 [t, y] = strangaut(y, t, T, h); 1 13 % plot numerical solution 14 i g u r e 15 p l o t 3 (y(1, :), y(, :), t, b- ) 16 x l a b e l ( y_1 ) 17 y l a b e l ( y_ ) 18 z l a b e l ( t, Rotation, ) 19 t i t l e ( numerical solution o the autonomized IVP via the strang splitting method ) Problem 8. Strang-Splitting For continuous dierentiable unction g : D R d R d consider the autonomous dierential equation ( ) ( ) ( ) u u g(v) ẏ = (y) with y =, (y) = :=. (8..1) v v g(u) (8.a) Give the method equations o the Strang-splitting 1-step method [NUMODE, Eq. (.5.3)] or (8..1) which is based on the decomposition ( ) ( ) g(v) (y) = (y) + 1 (y) := +. (8..) g(u) Solution: Exact evolution o the problems: ( ) ( u u + tg(v) Φ t 1 = v v ), Φ t ( ) ( u = v u v + tg(u) Problem Sheet 8 Page 3 Problem 8. ).

4 numerical solution o the autonomized IVP via the strang splitting method 1 8 t y y Figure 8.1: Numerical solution o the autonomized linear IVP using the Strang splitting method. Plugging this into the ormula [NUMODE, Eq. (.5.3)] rom the lecture gives the discrete evolution o the Strang-splitting 1-step method ( ) ( ) ( u Ψ h = Φ h/ Φ h 1 Φ h/ u u + hg(v + 1 = hg(u)) ) v v v + 1hg(u) + 1hg(u + hg(v + 1hg(u))). (8..3) (8.b) Is the 1-step method rom subproblem (8.a) an explicit or an implicit method? Solution: The method is explicit, because Ψ t can be evaluted without solving an equation. (8.c) What is the minimal order o this method in the general case? Solution: Order by [NUMODE, Thm..5.5] rom the lecture. (8.d) Implement a MATLAB-unction u = sped(g,u,t,n), that realizes N equidistant iterations o the Strang-splitting 1-step method or (8..1) with initial value ( u ) u on the time interval [, T ] using a minimal number o g( ) evaluations. The unction argument g is a The return value is a d (N + 1) matrix o the values o the u-component o the approximation y k, k =,..., N. Solution: Let ṽ := v + h g(u) then (8..3), turns into ( ) ( ) u1 u + hg(ṽ = ), (8..4) ṽ + hg(u 1 ) ṽ 1 Problem Sheet 8 Page 4 Problem 8.

5 which requires N evaluations o the unction g. See Listing 8.3. Listing 8.3: subproblem (8.d) 1 u n c t i o n u = sped(g,u,t,n) % unction sped (Strang Splitting) 3 % g : handle to right hand side unction 4 % u: initial value 5 % T : inal time 6 % N : number o timesteps 7 % return value: values or u-components h = T/N; 11 u = z e r o s ( l e n g t h (u),n+1); 1 u(:,1) = u; 13 v = u +.5*h*g(u(:,1)); o r k=1:n-1 16 u(:,k+1) = u(:,k) + h*g(v); 17 v = v + h*g(u(:,k+1)); 18 end 19 u(:,n+1) = u(:,n) + h*g(v); (8.e) Write a MATLAB-unction spedcirc.m that numerically solves the equation o motion or circular motion ẋ = y, ẏ = x or x() = 1, y() = using the method sped rom subproblem (8.d) with T = 6, N =. It should also plot the solution. Solution: In order to solve the circular motion using the method sped rom subproblem (8.d) we set u = (x, y) and v = (x, y), see Listing 8.4. For a plot o the solution see Figure 8.. Listing 8.4: subproblem (8.e) 1 % MATLAB script: circular movement with sped 3 % right hand side 4 g [-y();y(1)]; 5 6 % initial value 7 y = [1;]; 8 9 % Final time 1 T = 6; 11 1 % Number o timesteps Problem Sheet 8 Page 5 Problem 8.

6 sped: T = 6., N = y Figure 8.: Approximated trajectory o the solution in subproblem (8.e). x 13 N = ; y = sped(g,y,t,n); % Plot trajectory i g u r e ( name, spedcirc ); p l o t (y(1,:),y(,:), r. ); hold on; 1 p l o t (cos(:.1:*pi), s i n (:.1:*pi), g- ); p l o t ([ ],[ ], k- ); 3 p l o t ([ ],[ ], k- ); 4 x l a b e l ( {\b x}, ontsize,14); 5 y l a b e l ( {\b y}, ontsize,14); 6 t i t l e ( s p r i n t ( sped: T = %, N = %d,t,n)); 7 a x i s ([ ]); 8 9 p r i n t -depsc spedcirc.eps ; 3 31 i g u r e ( name, length deviation ); 3 p l o t ((:T/N:T), (y(1,:).ˆ+y(,:).ˆ)-1, r+ ); hold on; 34 x l a b e l ( {\b t}, ontsize,14); 35 y l a b e l ( {\b u_k ˆ-1}, ontsize,14); 36 t i t l e ( s p r i n t ( sped: T = %, N = %d,t,n)); p r i n t -depsc spedcirclen.eps ; Problem Sheet 8 Page 6 Problem 8.

7 (8.) Investigate numerically i the approximations u k rom sped in subproblem (8.e) are on a circle. Solution: The length is not preserved, as can be seen in Figure 8.3. One can also show this by perorming one step o the method with sped and subtracts the norm o the resulting vector by the norm o the initial vector ( 1 )..5 sped: T = 6., N =..15 u k Figure 8.3: Length variation in the numerical experiment rom subproblem (8.) t Problem 8.3 Stability o a Decomposed Method For the autonomous ODE ẏ = F(y) we will consider the ollowing decomposition o the right hand side, which we assume to be suiciently smooth: F(y) = DF(y )(y y ) + F(y) DF(y )(y y ). (8.3.1) }{{}}{{} =:(y) =:g(y) Taking y = y we deine the discrete evolution y 1 = Ψ h y o a single-step method as ollows: Compute ỹ by perorming one step o the explicit trapezoid rule with step-size h the initial value problem ẏ = g(y), y() = y. applied to Compute ŷ as the exact solution o the initial value problem ẏ = (y), y() = ỹ at the point t = h. Compute y 1 by applying the explicit trapezoid rule with step-size h problem ẏ = g(y), y() = ŷ. to the initial vlaue Problem Sheet 8 Page 7 Problem 8.3

8 (8.3a) Determine the stability unction o the method described above. HINT: To check your results, use the MATLAB unction S=stabn(z) (it is given as a pcode). It evaluates the stability unction at a given point z. Solution: To ind the stability unction, we apply the method to the model problem ẏ = λy =: F (y), y() = y. (8.3.) Because DF (y) = λ, the corresponding decomposition is given by g(y) = λy and (y) = λ(y y ). Thus, we receive ỹ = y + h 4 [g(y ) + g(y + h g(y ))] = y + h 4 [λy + λy ] = (1 + 1 λh)y ẏ = λ(y y ) d dt (y y ) = λ(y y ) y(h) y = (y() y )e λh ŷ = y + (ỹ y )e λh = (1 + 1 λheλh )y y 1 = ŷ + h 4 [g(ŷ) + g(ŷ + h g(ŷ))] = ŷ + h 4 [λy + λy ] = (1 + 1 λh + 1 λheλh )y The method s stability unction is now given as S(z) = z + 1 zez. (8.3b) Complete the MATLAB template plotstabdom which plots the boundary o the stability domain. Save the resulting plot as stabdom.eps. HINT: Use the MATLAB unction contour to draw the boundary. Take 5 and 5 as bounds or both axes. Solution: See Listing u n c t i o n plotstabdom % PLOTSTABDOM Listing 8.5: plotstabdom.m 3 % plots the boundary o the stability domain 4 % o the given method 5 % 6 % Input: 7 % none 8 % 9 % Output: 1 % none 11 1 % deine stability unction 13 S 1 + (1 + exp(z)).* z/; % create grid 16 [X, Y] = meshgrid( l i n s p a c e (-5, 5, 1)); % evaluate stability unction on grid 19 Z = abs(s(x + 1i*Y)); Problem Sheet 8 Page 8 Problem 8.3

9 Figure 8.4: Method s Stability Function 1 % plot stability domain contour(x, Y, Z, [1 1]); 3 4 % save plot 5 p r i n t -depsc stabdom.eps The stability domain is depicted in Figure 8.4 (8.3c) Show that the method described above has a convergence rate o at least. HINT: Notice that the previously described method can be written as a well known (and well studied) method. A theorem rom the lectures will then yield the statement. Solution: Let Φ and Φ g respectively be the exact evolutions to the autonomous dierential equations with right hand side and g respectively. Further, let Ψ g be the discrete evolution or ẏ = g(y) that is calculated by the explicit trapezoid rule. With this notation, the (discrete) evolution Ψ h o the given method can be written as Ψ h = Ψ h g Φ h Ψ h g. In other words, it is a Strang splitting method, in which the exact evolution Φ h g is replaced by Ψ h g. As we know, Strang splitting methods as well as the explicit trapezoid rule have consistency order, meaning, Φ h y = (Φ h g Φ h Φ h g )y + O(h 3 ), (8.3.3) Φ h gy = Ψ h gy + O(h 3 ) (8.3.4) holds, where Φ h denotes the exact evolution or the autonomous dierential equation ẏ = F(y). Problem Sheet 8 Page 9 Problem 8.3

10 As it ollows that Φ h i (y + y) = Φ h i y + O( y ), i {, g}, (8.3.5) Φ h y (8.3.3) = Φ h g ( Φ h (Φ h g y) ) + O(h 3 ) (8.3.4) = Φ h g ( Φ h (Ψ h g y + O(h 3 )) ) + O(h 3 ) (8.3.5) = Φ h g ( Φ h (Ψ h g y) + O(h 3 ) ) + O(h 3 ) (8.3.5) = Φ h g ( Φ h (Ψ h g y) ) + O(h 3 ) (8.3.4) = Ψ h g ( Φ h (Ψ h g y) ) + O(h 3 ) = Ψ h y + O(h 3 ). The consistency and thereby the convergence order o the method is thus. Published on 14 April 15. To be submitted by 1 April 15. Reerences [NUMODE] Lecture Slides or the course Numerical Methods or Ordinary Dierential Equations, SVN revision # Last modiied on April 1, 15 Problem Sheet 8 Page 1 Reerences