Lecture 6. Numerical Solution of Differential Equations B21/B1. Lecture 6. Lecture 6. Initial value problem. Initial value problem

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1 Lecture 6 Numerical Solution of Differential Equations B2/B Initial value problem y = f(x,y), x [x,x M ] y(x ) = y Professor Endre Süli Numerical Solution of Differential Equations p./45 Numerical Solution of Differential Equations p.2/45 Lecture 6 Lecture 6 Initial value problem Initial value problem y = f(x,y), x [x,x M ] y(x ) = y y = f(x,y), x [x,x M ] y(x ) = y General form of a linear multistep method: General form of a linear multistep method: α j y n+j = h β j f n+j f n+j = f(x n+j,y n+j ), α k, α + β Numerical Solution of Differential Equations p.2/45 Numerical Solution of Differential Equations p.2/45

2 Stiff problems: Van der Pol s oscillator Stiff problems: Van der Pol s oscillator y + µ(y 2 )y + y =, y() = 2, y () = with µ =,, 2, 5,, 2, 5. Solve on the interval [, 6]. y + µ(y 2 )y + y =, y() = 2, y () = with µ =,, 2, 5,, 2, 5. Solve on the interval [, 6]. y = y 2, y () = 2 y 2 = µ( y 2 )y 2 y, y 2 () = Numerical Solution of Differential Equations p.3/45 Numerical Solution of Differential Equations p.3/45 Result for µ = : y = y Result for µ = : y = y Numerical Solution of Differential Equations p.4/ Numerical Solution of Differential Equations p.5/45

3 Result for µ = 2: y = y Result for µ = 5: y = y Result for µ = : y = y Numerical Solution of Differential Equations p.6/ Result for µ = 2: y = y Numerical Solution of Differential Equations p.7/ Numerical Solution of Differential Equations p.8/ Numerical Solution of Differential Equations p.9/45

4 Result for µ = 5: y = y A stiff linear system 2.5 Let λ < and consider 2.5 y (λ )y λy =, y() =, y () = λ 2.5 Write y = y, y 2 = y, y = (y,y 2 ) T, y = Ay, y() = y..5 A = ( λ λ ) and y = ( λ 2 ) A stiff linear system Numerical Solution of Differential Equations p./45 A stiff linear system: λ = 45 Numerical Solution of Differential Equations p./45 det(a zi) = z 2 (λ )z λ 45 z =, y (x) = 2e x e λx, z 2 = λ y 2 (x) = 2e x λe λx y y Numerical Solution of Differential Equations p.2/45 5 Numerical Solution of Differential Equations p.3/45

5 Matlab code (for λ = 45) The simplest example linear.m function dy = lineq(x,y) lambda=-45; dy = zeros(2,); dy() = y(2); dy(2) = lambda * y()+(lambda-)*y(2); launchlin.m lambda=-45; [x,y] = ode3( linear,[ 2],[ -lambda-2]); plot(x,y(:,), kd-, x, y(:,2), kp- ) where Exact solution y = λy, y() = Re(λ) <. y(x) = e λx = e xre(λ) e ıxim(λ) Numerical Solution of Differential Equations p.4/45 Numerical Solution of Differential Equations p.5/45 The simplest example: y = λy, y() =, for λ = 45 Van der Pol s equation by Euler s method: h = u Numerical Solution of Differential Equations p.6/ t Numerical Solution of Differential Equations p.7/45

6 Van der Pol s equation by Euler s method: h =.84 Matlab code for van der Pol s eq: y + µ(y 2 )y + y = u x t Numerical Solution of Differential Equations p.8/45 What has gone wrong?... Absolute stability h = input( h? ); mu = 2; Xmax = 5; nmax = round(xmax/h); y = [.5 ]; yvec = [y; zeros(nmax,2)]; xvec = zeros(nmax+,); for i = :nmax yprime = [y(2) -mu*(y().ˆ2-)*y(2)-y()]; y = y + h*yprime; yvec(i+,:) = y; xvec(i+) = i*h; end plot(xvec,yvec(:,), linewidth,), grid Absolute stability xlabel( x ), ylabel( y ) Numerical Solution of Differential Equations p.9/45 Now suppose we apply α j y n+j = h β j f(x n+j,y n+j ) to this model problem (f(x,y) = λy with Re(λ) < ). Then, General solution (α j hβ j )y n+j = y n = p s (n)zs n s (α j λhβ j )y n+j = Let us write h = λh. Clearly, Re( h) <. where z s is a zero of the stability polynomial π(z,h) ρ(z) hσ(z) = (α j hβ j )z j Numerical Solution of Differential Equations p.2/45 Numerical Solution of Differential Equations p.2/45

7 Absolute stability Absolute stability Now, Therefore, we want For this to hold, we need lim y(x) = x lim y n = n A linear multistep method is called absolutely stable in an open set R A of the complex plane, if for all h R A all roots z s = z s ( h), s =,...,k, of π(z, h) satisfy z s <. The set R A is called the region of absolute stability. z s < for all s =,...,k. Numerical Solution of Differential Equations p.22/45 Numerical Solution of Differential Equations p.23/45 Exercise Find the region of absolute stability of: Euler s method Exercise Find the region of absolute stability of: Euler s method The implicit Euler method Numerical Solution of Differential Equations p.24/45 Numerical Solution of Differential Equations p.24/45

8 Region of absolute stability of Euler s method Region of absolute stability of Euler s method y n+ y n = hf(x n,y n ) Apply to y = λy, y() =, where Re(λ) <. y n+ y n = hf(x n,y n ) Apply to y = λy, y() =, where Re(λ) <. ρ(z) = z, σ(z) = Numerical Solution of Differential Equations p.25/45 Numerical Solution of Differential Equations p.25/45 Region of absolute stability of Euler s method Region of absolute stability of Euler s method y n+ y n = hf(x n,y n ) Apply to y = λy, y() =, where Re(λ) <. ρ(z) = z, σ(z) = y n+ y n = hf(x n,y n ) Apply to y = λy, y() =, where Re(λ) <. ρ(z) = z, σ(z) = π(z, h) = z h π(z, h) = z h z = + h Numerical Solution of Differential Equations p.25/45 Numerical Solution of Differential Equations p.25/45

9 Region of absolute stability of Euler s method Region of absolute stability: interior of circle y n+ y n = hf(x n,y n ) Apply to y = λy, y() =, where Re(λ) <. ρ(z) = z, σ(z) = π(z, h) = z h z = + h - z < iff + h < R A = { h C : + h < } Numerical Solution of Differential Equations p.25/45 Numerical Solution of Differential Equations p.26/45 y = λy, y() =, λ =, h =. y = λy, y() =, λ =, h = y.5 y t Numerical Solution of Differential Equations p.27/ t Numerical Solution of Differential Equations p.28/45

10 y = λy, y() =, λ =, h =. y = λy, y() =, λ =, h = y.5 y t Numerical Solution of Differential Equations p.29/ t Numerical Solution of Differential Equations p.3/45 y = λy, y() =, λ =, h =.5 y = λy, y() =, λ =, h = y y t Numerical Solution of Differential Equations p.3/ t Numerical Solution of Differential Equations p.32/45

11 Matlab code for y = λy, y() =, by Euler s method Absolute stability of the implicit Euler method h=input( h? ); lambda = -; Xmax = 3; nmax = round(xmax/h); y = ; yvec = [y; zeros(nmax,)]; xvec = zeros(nmax+,); for i = :nmax yprime = lambda*y(); y = y + h*yprime; yvec(i+,:) = y; xvec(i+) = i*h; end plot(xvec,yvec(:,), k-, linewidth,2), grid xlabel( x ), Absolute stability ylabel( y ) of the implicit Euler method Numerical Solution of Differential Equations p.33/45 y n+ y n = hf(x n+,y n+ ) Apply to y = λy, y() =, where Re(λ) <. Absolute stability of the implicit Euler method Numerical Solution of Differential Equations p.34/45 y n+ y n = hf(x n+,y n+ ) Apply to y = λy, y() =, where Re(λ) <. y n+ y n = hf(x n+,y n+ ) Apply to y = λy, y() =, where Re(λ) <. ρ(z) = z, σ(z) = z ρ(z) = z, σ(z) = z π(z, h) = z hz Numerical Solution of Differential Equations p.34/45 Numerical Solution of Differential Equations p.34/45

12 Absolute stability of the implicit Euler method Absolute stability of the implicit Euler method y n+ y n = hf(x n+,y n+ ) Apply to y = λy, y() =, where Re(λ) <. y n+ y n = hf(x n+,y n+ ) Apply to y = λy, y() =, where Re(λ) <. ρ(z) = z, σ(z) = z ρ(z) = z, σ(z) = z π(z, h) = z hz z = h π(z, h) = z hz z = h R A = { h C : h > } Region of absolute stability: exterior of circle Numerical Solution of Differential Equations p.34/45 A stability Numerical Solution of Differential Equations p.34/45 A linear multistep method is said to be A stable if its region of absolute stability, R A, contains the whole of the open left-hand complex half plane, Re( h) <. Example: The implicit Euler method is A stable. Numerical Solution of Differential Equations p.35/45 Numerical Solution of Differential Equations p.36/45

13 The Dahlquist Barrier Theorem The Dahlquist Barrier Theorem No explicit linear multistep method is A stable. No explicit linear multistep method is A stable. The order of an A stable implicit linear multistep method is 2. Numerical Solution of Differential Equations p.37/45 The Dahlquist Barrier Theorem Numerical Solution of Differential Equations p.37/45 Stepsize control: The Milne device No explicit linear multistep method is A stable. The order of an A stable implicit linear multistep method is 2. The second order A stable linear multistep method with smallest error constant is the trapezium rule method. Suppose we are using the pth order linear k-step method α j y n+j = h β j f(x n+j,y n+j ) How to choose h appropriately? Numerical Solution of Differential Equations p.37/45 Numerical Solution of Differential Equations p.38/45

14 Stepsize control: The Milne device The Milne device Suppose we are using the pth order linear k-step method α j y n+j = h β j f(x n+j,y n+j ) How to choose h appropriately? If y n,y n+,...,y n+k are error free, then y(x n+k ) y n+k = ch p+ y (p+) (x n+k ) + O(h p+2 ) h c is called the local error constant. [Proof at the back.] o Consider another pth order method: α j ỹ n+j = h β j f(x n+j,ỹ n+j ) Numerical Solution of Differential Equations p.38/45 Numerical Solution of Differential Equations p.39/45 The Milne device The Milne device If y n,y n+,...,y n+k are error free, then y(x n+k ) y n+k = ch p+ y (p+) (x n+k ) + O(h p+2 ) h c is called the local error constant. [Proof at the back.] o Similarly, Then, Hence, ỹ n+k y n+k (c c)h p+ y (p+) (x n+k ) y(x n+k ) y n+k c c c (ỹ n+k y n+k ) y(x n+k ) ỹ n+k = ch p+ y (p+) (x n+k ) + O(h p+2 ) h Suppose c c. Numerical Solution of Differential Equations p.39/45 Numerical Solution of Differential Equations p.4/45

15 Then, Hence, The Milne device ỹ n+k y n+k (c c)h p+ y (p+) (x n+k ) y(x n+k ) y n+k c c c (ỹ n+k y n+k ) The Milne device: local refinement criteria Error control per step Given a fixed tolerance δ, require that κ δ Define local refinement indicator: κ = c c c ỹ n+k y n+k Numerical Solution of Differential Equations p.4/45 Numerical Solution of Differential Equations p.4/45 The Milne device: local refinement criteria The Milne device: the adaptive algorithm Error control per step Given a fixed tolerance δ, require that. Set h κ δ Error control unit per step Given a fixed tolerance δ, require that κ hδ Numerical Solution of Differential Equations p.4/45

16 The Milne device: the adaptive algorithm The Milne device: the adaptive algorithm. Set h. Set h 3. Is κ hδ? The Milne device: the adaptive algorithm The Milne device: the adaptive algorithm. Set h 3. Is κ hδ? NO: halve h, remesh and GOTO 2. Set h 3. Is κ hδ? NO: halve h, remesh and GOTO 2 YES: GOTO 4

17 The Milne device: the adaptive algorithm The Milne device: the adaptive algorithm. Set h 3. Is κ hδ? NO: halve h, remesh and GOTO 2 YES: GOTO 4 4. Is x X M?. Set h 3. Is κ hδ? NO: halve h, remesh and GOTO 2 YES: GOTO 4 4. Is x X M? YES: STOP The Milne device: the adaptive algorithm The Milne device: the adaptive algorithm. Set h 3. Is κ hδ? NO: halve h, remesh and GOTO 2 YES: GOTO 4 4. Is x X M? YES: STOP NO: GOTO 5. Set h 3. Is κ hδ? NO: halve h, remesh and GOTO 2 YES: GOTO 4 4. Is x X M? YES: STOP NO: GOTO 5 5. Is κ hδ?

18 The Milne device: the adaptive algorithm The Milne device: the adaptive algorithm. Set h 3. Is κ hδ? NO: halve h, remesh and GOTO 2 YES: GOTO 4 4. Is x X M? YES: STOP NO: GOTO 5 5. Is κ hδ? YES: double h, advance x, remesh, GOTO 2. Set h 3. Is κ hδ? NO: halve h, remesh and GOTO 2 YES: GOTO 4 4. Is x X M? YES: STOP NO: GOTO 5 5. Is κ hδ? YES: double h, advance x, remesh, GOTO 2 NO: advance x, GOTO 2 Proof Proof α k y n+k hβ k f(x n+k,y n+k ) k + k α j y n+j h β j f(x n+j,y n+j ) = By the definition of the truncation error T n, and writing B = k β j, α k y(x n+k ) hβ k f(x n+k,y(x n+k )) k + k α j y(x n+j ) h β j f(x n+j,y(x n+j )) = B ht n Since y n,y n+,...,y n+k are error free, α k [y(x n+k ) y n+k ] hβ k [f(x n+k,y(x n+k )) f(x n+k,y n+k )] = B ht n α k [y(x n+k ) y n+k ] hβ k f y (x n+k,η n+k ) [y(x n+k ) y n+k ] = B ht n [ ] f α k hβ k y (x n+k,η n+k ) (y(x n+k ) y n+k ) = B ht n Numerical Solution of Differential Equations p.43/45 Numerical Solution of Differential Equations p.44/45

19 Proof As h, Therefore, Recalling that α k hβ k f y (x n+k,η n+k ) α k y(x n+k ) y n+k = Const. ht n T n = Const.h p y (p+) (x n+k ) + O(h p+ ) we deduce the desired result. o Numerical Solution of Differential Equations p.45/45