Runge-Kutta and Collocation Methods Florian Landis Geometrical Numetric Integration p.1
Overview Define Runge-Kutta methods. Introduce collocation methods. Identify collocation methods as Runge-Kutta methods. Find conditions to determine, of what order collocation methods are. Geometrical Numetric Integration p.2
Introduction General Goal: Find approximation to the solutions of using one step methods. ẏ(t) = f(t, y), y(t ) = y 3 Examples of one step methods (step size h = 1) for the Riccati equation t y = y 2 + t 2 : 1.5 Explicit Euler Rule 1.5 Explicit Trapezoidal Rule 1.5 Explicit 3 stage Runge Kutta method 1 1 1 y y y.5.5.5 1 t 1 t c2 c3 1 t Geometrical Numetric Integration p.3
Runga-Kutta Method Definition 1 (Runge-Kutta) Let b i, a ij (i,j = 1,...,s) be real numbers and let c i = s a ij. An s-stage Runge-Kutta method is given by ( k i = f t + c i h, y + h y 1 = y + h b i k i. a ij k j ), i = 1,...,s (1) Distinguish: explicit Runge-Kutta a ij = for j i implicit Runge-Kutta full matrix (a ij ) of non-zero coefficients allowed Implicit function theorem: for h small enough, (1) has a locally unique solution close to k i f(t,y ). Geometrical Numetric Integration p.4
Butcher Diagram The coefficients of the Runge-Kutta method are usually displayed in a Butcher diagram: c 1 a 11... a 1s... c s a s1... a ss b 1... b s Example for explicit Runge-Kutta: 1.5 Runge Kutte method for Riccati equation t y=y 2 +t 2 b 1 b 2 b 3.5.5.7.3.4.2.25.55 y 1.5 a 21 a 31 a 32 c2 c3 1 t Geometrical Numetric Integration p.5
Order of the Runge-Kutta method A general one-step method has order p, if By the Taylor expansions y 1 y(t + h) = O(h p+1 ) ash. y(t + h) = y(t ) + h f(t, y(t )) + 1 2 h2 d dt f(t, y(t)) t=t +... d ( ) ] y 1 = y + h b i [f(t, y ) + h dh f h= t + c i h, y + h a ij k j +... i=1 of y and y 1 of the Runge-Kutta method, one obtains the following conditions for the coefficients: i b i = 1 for order 1, i b ic i = 1/2 for order 2, i b ic 2 i = 1/3 and i,j b ia ij c j = 1/6 for order 3. Geometrical Numetric Integration p.6
The Collocation Method Definition 2 (Collocation Method) Let c 1,...,c s be distinct real numbers (usually c i 1). The collocation polynomial u(t) is a polynomial of degree s satisfying u(t ) = y (2) u(t + c i h) = f(t + c i h, u(t + c i h)), i = 1,...,s, (3) and the numerical solution of the collocation method is defined by Scetch of Collocation Polynomial of degree 3 y 1 = u(t + h). y 1.5 y 1 1.5 u y 1 t c 1 c 2 c 3 Geometrical Numetric Integration p.7
The Collocation Method Theorem 1 (Guillou & Soulé 1969, Wright 197) The collocation method for c 1,...,c s is equivalent to the s-stage Runge-Kutta method with coefficients a ij = ci l j (τ) dτ, b i = 1 l i (τ) dτ, where l i (τ) is the Lagrange polynomial l i (τ) = l i (τ c l)/(c i c l ). Moreover: u(t + τh) = y + h τ k j l j (σ) dσ. Thus, the existence of the collocation polynomial depends on the existence of the k i (given for h ). Geometrical Numetric Integration p.8
Proof of Theorem 1 Proof. Let u(t) be the collocation polynomial and define k i := u(t + c i h). By the Lagrange interpolation formula we have u(t + τh) = s k j l j (τ), and by integration we get u(t + c i h) = y + h ci k j l j (τ) dτ. Inserted into the definition of the collocation polynomial u(t + c i h) = f(t + c i h, u(t + c i h)), this gives the first formula of the Runge-Kutta equation ( k i = f t + c i h, y + h a ij k j ). Integration from to 1 yields y 1 = y + h s b ik i. Geometrical Numetric Integration p.9
Collocation Coefficients If a Runge-Kutta methods corresponds to a collocation method of order s, a ij = ci l j (τ) dτ, b i = 1 l i (τ) dτ, leads to: C(q = s) : B(p = s) : a ij c k 1 j = ck i k, i, k = 1,...,q b i c k 1 i = 1, k = 1,...,p k since τ k 1 = s j 1 ck 1 j l j (τ) for k = 1,...,s. Note: B(p) y + s i=1 b if(t + hc i ) approximates the solution to ẏ = f(t), y(t ) = y with order p. Geometrical Numetric Integration p.1
Order of The Collocation Method Lemma 2 The collocation polynomial u(t) is an approximation of order s to the exact solution of ẏ = f(t,y), y(t ) = y on the whole interval, i.e., u(t) y(t) C h s+1 for t [t, t + h] and for sufficiently small h. Moreover, the derivatives of u(t) satisfy for t [t,t + h] u (k) (t) y (k) (t) C h s+1 k for k =,...,s. Geometrical Numetric Integration p.11
Proof of Lemma 2 y u(t + τh) = ẏ(t + τh) = f ( t + c i h, u(t + c i h) ) l j (τ), f ( t + c i h, y(t + c i h) ) t l j (τ) + h s E(τ, h) y t + τh h s E y Σ f i l i u t E(τ, h) 2 max t [t,t +h] y (s+1) (t) s! Integrating the difference of the above two equations gives y(t + τh) u(t + τh) = h τ f i i=1 l i (σ) dσ + h s+1 τ E(σ, h) dσ with f i = f ( t + c i h,y(t + c i h) ) f ( t + c i h,u(t + c i h) ). Geometrical Numetric Integration p.12
Proof of Lemma 2 Using a Lipschitz condition for f(t, y) on the Integral y(t + τh) u(t + τh) = h τ f i i=1 l i (σ) dτ + h s+1 τ E(σ, h) dσ yields max y(t) u(t) h C L max y(t) u(t) + Const. t [t,t +h] t [t,t +h] hs+1, implying u(t) y(t) C h s+1 for sufficiently small h >. Geometrical Numetric Integration p.13
Superconvergence Theorem 3 (Superconvergence) If the condition B(p) holds for some p s, then the collocation method has order p. This means that the collocation method has the same order as the underlying quadrature formula. B(p) : b i c k 1 i = 1, k = 1,...,p k Note: B(p) cannot be met for p > 2s. Geometrical Numetric Integration p.14
Proof of Superconvergence Proof. We consider the collocation polynomial u(t) as the solution of a perturbed differential equation u = f(t, u) + δ(t) with defect δ(t) := u(t) f ( t,u(t) ). Subtracting ẏ(t) = f(t,y) from the above we get after linearization that u(t) ẏ(t) }{{} E(t) = f ( ) ( ) t, y(t) u(t) y(t) y }{{} E(t) where, for t t t + h, the remainder r(t) is of size O ( u(t) y(t) 2) = O(h 2s+2 ) by lemma 2. +δ(t) + r(t), Geometrical Numetric Integration p.15
Conclusion Collocation methods with polynomials of degree s are equivalent to s-stage Runge-Kutta methods: a ij = ci l j (τ) dτ, b i = 1 l i (τ) dτ, Collocation polynomials of degrees s lead to collocation methods of order s or better: If B(p) is met for p > s, the corresponding collocation method is of order p. B(p) : b i c k 1 i = 1, k = 1,...,p. k Geometrical Numetric Integration p.16
Geometrical Numetric Integration p.17
Proof of Lemma 2 The second statement follows from the first one: Taking the kth derivative of y(t + τh) u(t + τh) = h τ f i i=1 l i (σ) dτ + h s+1 τ E(σ, h) dσ gives h k( y (k) (t + τh) u (k) (t + τh) ) = h i=1 f i l (k 1) i (τ) + h s+1 E (k 1) (τ, h). With E (k 1) (τ, h) max t [t,t +h] y (s+1) (t) (s k + 1)! and a Lipschitz condition for f(t,y), u (k) y (k) C h s+1 k follows. Geometrical Numetric Integration p.18
Variation of Constants Formula For homogeneous systems of linear equations ẏ(t) = A(t)y(t) with initial condition y(t ) = y, the solution can be written as y(t) = R(t,t )y Ṙ(t,s) = A(t)R(t,s). Using this resolvent of the homogeneous differential system, the solution to inhomogeneous problems ẏ(t) = A(t)y(t) + f(t) can be found with the variation of constants formula: y(t) = R(t,t )y + t t R(t,s)f(s)ds. Geometrical Numetric Integration p.19
Proof of Superconvergence With Ṙ(t, s) = f(t, y(t))r(t, s) y The variation of constants formula then yields y 1 y(t + h) = E(t + h) = t +h t ( ) R(t + h, s) δ(s) + r(s) ds as the solution of The contribution of r(t): E(t + h) = f ( ) t, y(t) E(t) + δ(t) + r(t). y r(t) O(h 2s+2 ) t +h t R(t + h, s)r(s) ds O(h 2s+3 ) Geometrical Numetric Integration p.2
Proof of Superconvergence The main idea now is to apply the quadrature formula (b i,c i ) s i=1 integral of g(s) = R(t + h,s)δ(s): to the t +h t g(s) ds = b i g(t + hc i ) + quadrature Error. i=1 From δ(s) t +c i h = follows s i=1 b ig(t + hc i ) =. Thus, t +h t p+1 p g(s) ds = quadrature Error h s p g(s). p s p g(s) is bounded independently of h by Lemma 2. Therefore E(t + h) = t +h t ( ) R(t + h, s) δ(s) + r(s) ds O(h p+1 ). Geometrical Numetric Integration p.21