Numerical Methods for Differential Equations

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1 Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Linear Multistep methods Gustaf Söderlind and Carmen Arévalo Numerical Analysis, Lund University Textbooks: A First Course in the Numerical Analysis of Differential Equations, by Arieh Iserles and Introduction to Mathematical Modelling with Differential Equations, by Lennart Edsberg c Gustaf Söderlind, Numerical Analysis, Mathematical Sciences, Lund University, Numerical Methods for Differential Equations p. 1/63

2 Chapter 2: contents Solving nonlinear equations Fixed points Newton s method Quadrature Runge Kutta methods Embedded RK methods and adaptivity Implicit Runge Kutta methods Stability and the stability function Linear multistep methods Numerical Methods for Differential Equations p. 2/63

3 1. Solving nonlinear equations f(x) = 0 We can have a single equation x cos(x) = 0 or a system 4x 2 y 2 = 0 4xy 2 x = 1 Nonlinear equations may have no solution; one solution; any finite number of solutions; infinitely many solutions Numerical Methods for Differential Equations p. 3/63

4 Iteration and convergence Nonlinear equations are solved iteratively. One computes a sequence {x [k] } of approximations to the root x For f(x ) = 0, let e [k] = x [k] x Definition The method converges if lim k e [k] = 0 Definition The convergence is linear if e [k+1] c e [k] with 0 < c < 1 quadratic if e [k+1] c e [k] p with p = 2 superlinear if p > 1; cubic if p = 3, etc. Numerical Methods for Differential Equations p. 4/63

5 2. Fixed points Definition x is called a fixed point of the function g if x = g(x) Definition A function g is called contractive if g(x) g(y) L[g] x y with L[g] < 1 for all x,y in the domain of g Numerical Methods for Differential Equations p. 5/63

6 Fixed Point Theorem Theorem Assume that g is continuously differentiable on the compact interval I, i.e., g C 1 (I) If g : I I there exists an x I such that x = g(x ) If in addition L[g] < 1 on I, then x is unique, and the iteration x n+1 = g(x n ) converges to the fixed point x for all x 0 I Note Both conditions are absolutely essential! Numerical Methods for Differential Equations p. 6/63

7 Fixed Point Theorem. Existence and uniqueness Left Center Right No condition satisfied: maybe no x First condition satisfied: maybe multiple x Both conditions satisfied: unique x Numerical Methods for Differential Equations p. 7/63

8 Fixed point iteration. x = e x ; g(x) = e x x [k+1] = e x[k] ; x [0] = 0 g : [0,1] [0,1]; g (x) = e x < 1 for x > Numerical Methods for Differential Equations p. 8/63

9 Error estimation in fixed point iteration Assume g (x) L < 1 for all x I. x [k+1] = g(x [k] ) x = g(x ) By the mean value theorem, x [k+1] x = g(x [k] ) g(x ) = g(x [k] ) g(x [k+1] ) + g(x [k+1] ) g(x ) x [k+1] x L x [k] x [k+1] + L x [k+1] x Numerical Methods for Differential Equations p. 9/63

10 Error estimation... (1 L) x [k+1] x L x [k] x [k+1] Error estimate (computable error bound) x [k+1] x L 1 L x[k] x [k+1] Theorem If L[g] < 1, then the error in fixed point iteration is bounded by x [k+1] x L[g] 1 L[g] x[k] x [k+1] Numerical Methods for Differential Equations p. 10/63

11 Example: The trapezoidal rule Approximate the solution to y = y 2 cos t, y(0) = 1/2, t [0,8π]. Taking 96 steps, solve nonlinear equations with one (left) and four (right) fixed point iterations. Graphs show absolute error Numerical Methods for Differential Equations p. 11/63

12 3. Newton s method Newton s method solves f(x) = 0 using repeated linearizations. Linearize at the point (x [k],f(x [k] ))! Numerical Methods for Differential Equations p. 12/63

13 Straight line equation Newton s method... y f(x [k] ) = f (x [k] ) (x x [k] ) Define x = x [k+1] y = 0, so that f(x [k] ) = f (x [k] ) (x [k+1] x [k] ) Solve for x = x [k+1], to get Newton s method x [k+1] = x [k] f(x[k] ) f (x [k] ) Numerical Methods for Differential Equations p. 13/63

14 Newton s method. Alternative derivation Expand f(x [k+1] ) in a Taylor series around x [k] f(x [k+1] ) = f(x [k] + (x [k+1] x [k] )) f(x [k] ) + f (x [k] ) (x [k+1] x [k] ) := 0 Newton s method x [k+1] = x [k] f(x[k] ) f (x [k] ) Numerical Methods for Differential Equations p. 14/63

15 Newton s method. Alternative derivation... This holds also when f is vector valued (equation systems) f(x [k] ) + f (x [k] ) (x [k+1] x [k] ) := 0 x [k+1] = x [k] ( f (x [k] ) ) 1 f(x [k] ) Definition f (x [k] ) is the Jacobian matrix of f, defined by f (x) = { fi x j } Numerical Methods for Differential Equations p. 15/63

16 Newton s method. Convergence Write Newton s method as a fixed point iteration with iteration function g(x) := x f(x)/f (x) x [k+1] = g(x [k] ) Note Newton s method converges fast if f (x ) 0, as g (x ) = f(x )f (x )/f (x ) 2 = 0 Expand g(x) in a Taylor series around x g(x [k] ) g(x ) g (x )(x [k] x ) + g (x ) (x [k] x ) 2 2 x [k+1] x g (x ) (x [k] x ) 2 2 Numerical Methods for Differential Equations p. 16/63

17 Newton s method. Convergence... Define the error by ε [k] = x [k] x, then ε [k+1] ( ε [k]) 2 Newton s method is quadratically convergent! Fixed point iterations are typically only linearly convergent ε [k+1] ε [k] Numerical Methods for Differential Equations p. 17/63

18 Implicit Euler. Newton vs Fixed point As y n+1 = y n + hf(y n+1 ) we need to solve an equation y = hf(y) + ψ Note All implict methods lead to an equation of this form! Theorem Fixed point iterations converge if L[hf] < 1, restricting the step size to h < 1/L[f]! Stiff equations have L[hf] 1 so fixed point iterations will not converge; it is necessary to use Newton s method! Numerical Methods for Differential Equations p. 18/63

19 Convergence order and rate Definition The convergence order is p with (asymptotic) error constant C p, if 0 < lim k ε [k+1] ε [k] p = C p < Special cases: p = 1 p = 2 Linear convergence Example Fixed point iteration C p = g (x ) Quadratic convergence Example Newton iteration C p = f (x ) 2f (x ) Numerical Methods for Differential Equations p. 19/63

20 4. Quadrature (integration) formulas Numerical quadrature is the approximation of definite integrals I(f) = b a f(x)dx We substitute the infinite sum by a finite sum: the integrand is sampled at a finite number of points I(f) = n w i f(x i ) + R n i=1 R n = I(f) n i=1 w if(x i ) is the quadrature error. Numerical method n I(f) w i f(x i ) i=1 Numerical Methods for Differential Equations p. 20/63

21 5. Runge Kutta methods To solve the IVP y = f(t,y), t t 0, y(t 0 ) = y 0 we use a quadrature formula to approximate the integral in y(t n+1 ) = y(t n ) + y(t n+1 ) y(t n ) + h tn+1 t n f(τ,y(τ))dτ s b j f(t n + c j h,y(t n + c j h)) j=1 Let {Y j } denote the numerical approximations to {y(t n + c j h)} A Runge Kutta method then has the form y n+1 = y n + h s b j f(t n + c j h,y j ) j=1 Numerical Methods for Differential Equations p. 21/63

22 Stage values and stage derivatives The vectors Y j are called stage values The vectors Y j = f(t n + c j h,y j ) are called stage derivatives Stage values and stage derivatives are related through Y i = y n + h i 1 j=1 a i,j Y j and the method advances one step through y n+1 = y n + h s j=1 b j Y j Numerical Methods for Differential Equations p. 22/63

23 The RK matrix, weights, nodes and stages A = {a ij } is the RK matrix, b = [b 1 b 2 b s ] T is the weight vector, c = [c 1 c 2 c s ] T are the nodes, and the method has s stages The Butcher tableau of an explicit RK method is c 2 a c s a s1 a s2 0 or c A b T b 1 b 2 b s Simplifying assumption: c i = i a i,j i = 2,3,...,s j=1 Numerical Methods for Differential Equations p. 23/63

24 Simplest case: a 2-stage ERK Y 1 = f(t n,y n ) Y 2 = f(t n + c 2 h,y n + ha 21 Y 1) y n+1 = y n + h[b 1 Y 1 + b 2 Y 2] c 2 a 21 0 b 1 b 2 c 2 = a 21 Numerical Methods for Differential Equations p. 24/63

25 2-stage ERK... Using Y 1 = f(t n,y n ), expand in Taylor series around t n,y n : Y 2 = f(t n + c 2 h,y n + ha 21 f(t n,y n )) = f + h [c 2 f t + a 21 f y f] + O(h 2 ) Numerical Methods for Differential Equations p. 25/63

26 2-stage ERK... Using Y 1 = f(t n,y n ), expand in Taylor series around t n,y n : Y 2 = f(t n + c 2 h,y n + ha 21 f(t n,y n )) = f + h [c 2 f t + a 21 f y f] + O(h 2 ) Inserting into y n+1 = y n + h[b 1 Y 1 + b 2Y ], we get y n+1 = y n + h(b 1 + b 2 )f + h 2 b 2 (c 2 f t + a 21 f y f) + O(h 3 ) 2 Numerical Methods for Differential Equations p. 25/63

27 2-stage ERK... Using Y 1 = f(t n,y n ), expand in Taylor series around t n,y n : Y 2 = f(t n + c 2 h,y n + ha 21 f(t n,y n )) = f + h [c 2 f t + a 21 f y f] + O(h 2 ) Inserting into y n+1 = y n + h[b 1 Y 1 + b 2Y ], we get y n+1 = y n + h(b 1 + b 2 )f + h 2 b 2 (c 2 f t + a 21 f y f) + O(h 3 ) Taylor expansion of the exact solution: y y = f = f t + f y y = f t + f y f y(t + h) = y + hf h2 (f t + f y f) + O(h 3 ) 2 Numerical Methods for Differential Equations p. 25/63

28 2-stage ERK... Matching terms in y n+1 = y n + h(b 1 + b 2 )f + h 2 b 2 (c 2 f t + a 21 f y f) + O(h 3 ) y(t + h) = y + hf h2 (f t + f y f) + O(h 3 ) and taking c 2 = a 21 we get the conditions for order 2: b 1 + b 2 = 1 (consistency) b 2 c 2 = 1/2 Note All consistent RK methods are convergent! Second order, 2-stage ERK methods have a Butcher tableau b 1 2b 0 1 b b Numerical Methods for Differential Equations p. 26/63

29 Example 1. The modified Euler method Put b = 1 to get Y 1 = f(t n,y n ) Y 2 = f(t n + h/2,y n + hy 1/2) y n+1 = y n + hy 2 Second order explicit Runge Kutta (ERK) method Numerical Methods for Differential Equations p. 27/63

30 Example 2. Heun s method Put b = 1/2 to get /2 1/2 Y 1 = f(t n,y n ) Y 2 = f(t n + h,y n + hy 1) y n+1 = y n + h (Y 1 + Y 2)/2 Second order ERK, compare to the trapezoidal rule! Numerical Methods for Differential Equations p. 28/63

31 Third order 3-stage ERK Conditions for 3rd order: a 21 = c 2 ; a 31 + a 32 = c 3 b 1 + b 2 + b 3 = 1 b 2 c 2 + b 3 c 3 = 1/2 b 2 c b 3 c 2 3 = 1/3 b 3 a 32 c 2 = 1/6 Classical RK /2 1/ /6 2/3 1/6 Nyström scheme /3 2/ /3 0 2/3 0 1/4 3/8 3/8 Numerical Methods for Differential Equations p. 29/63

32 Exercise Construct the Butcher tableau for the 3-stage Heun method. Y 1 = f(t n,y n ) Y 2 = f(t n + h/3,y n + hy 1/3) Y 3 = f(t n + 2h/3,y n + 2hY 2/3) y n+1 = y n + h (Y 1 + 3Y 3)/4 Is it of order 3? Numerical Methods for Differential Equations p. 30/63

33 Classic RK4: 4th order, 4-stage ERK The original RK method (1895): Y 1 = f(t n,y n ) Y 2 = f(t n + h/2,y n + hy 1 /2) Y 3 = f(t n + h/2,y n + hy 2/2) Y 4 = f(t n + h,y n + hy 3) y n+1 = y n + h 6 (Y 1 + 2Y 2 + 2Y 3 + Y 4). Numerical Methods for Differential Equations p. 31/63

34 Classic 4th order RK4... Butcher tableau 0 1/2 1/2 1/2 0 1/ /6 1/3 1/3 1/6 s-stage ERK methods of order p = s exist only for s 4 (i.e. there is no 5-stage ERK of order 5) Numerical Methods for Differential Equations p. 32/63

35 Order conditions An s-stage ERK method has s + s(s 1)/2 coefficients to choose, but the order conditions are many Numerical Methods for Differential Equations p. 33/63

36 Order conditions An s-stage ERK method has s + s(s 1)/2 coefficients to choose, but the order conditions are many Number of coefficients stages s # coefficients Numerical Methods for Differential Equations p. 33/63

37 Order conditions An s-stage ERK method has s + s(s 1)/2 coefficients to choose, but the order conditions are many Number of coefficients stages s # coefficients Number of order conditions order p # conditions Numerical Methods for Differential Equations p. 33/63

38 Order conditions An s-stage ERK method has s + s(s 1)/2 coefficients to choose, but the order conditions are many Number of coefficients stages s # coefficients Number of order conditions order p # conditions Maximum order, min stages order p min stages Numerical Methods for Differential Equations p. 33/63

39 6. Embedded RK methods Two methods in one Butcher tableau (RK34) Y 1 = f(t n,y n ) Y 2 = f(t n + h/2,y n + hy 1/2) Y 3 = f(t n + h/2,y n + hy 2/2) Z 3 = f(t n + h,y n hy 1 + 2hY 2) Y 4 = f(t n + h,y n + hy 3) y n+1 = y n + h 6 (Y 1 + 2Y 2 + 2Y 3 + Y 4) order 4 z n+1 = y n + h 6 (Y 1 + 4Y 2 + Z 3) order 3 The difference y n+1 z n+1 can be used as an error estimate Numerical Methods for Differential Equations p. 34/63

40 Adaptive RK methods Example Use an embedded pair, e.g. RK34 Local error estimate r n+1 := y n+1 z n+1 = O(h 4 ) Adjust the step size h so that the local error estimate equals a prescribed tolerance TOL Simplest step size change scheme h n+1 = ( TOL r n+1 ) 1/p h n makes r n TOL Adaptivity using local error control Numerical Methods for Differential Equations p. 35/63

41 7. Implicit Runge Kutta methods (IRK) Y 1 Y 2 Y s y n+1 = y n + h = y n + h. = y n + h = y n + h s a 1,j f(t n + c j h,y j ) j=1 s a 2,j f(t n + c j h,y j ) j=1 s a s,j f(t n + c j h,y j ) j=1 s b j f(t n + c j h,y j ) j=1 Numerical Methods for Differential Equations p. 36/63

42 Implicit Runge Kutta methods... In stage value stage derivative form Y i = y n + h s j=1 a i,j Y j Y j = f(t n + c j h,y j ) y n+1 = y n + h s j=1 b j Y j Numerical Methods for Differential Equations p. 37/63

43 1-stage IRK method Implicit Euler (order 1) Implicit midpoint method (order 2) Y 1 = f(t n + c 1 h,y n + ha 11 Y 1) y n+1 = y n + hb 1 Y 1 may be written as 1/2 1/2 y n+1 = y n + hb 1 f(t n + c 1 h,( b 1 a 11 b 1 y n + a 11 b 1 y n+1 )) Taylor expansion: y n+1 = y + hb 1 f(t n + c 1 h,y + a 11 hf + a 11 h 2 b 1 b 1 2 (f t + f y f) + O(h 2 )) = y + hb 1 hf + h 2 (b 1 c 1 f t + a 11 f y f) + O(h 3 ) 1 Numerical Methods for Differential Equations p. 38/63

44 Taylor expansions for 1-stage IRK y(t n+1 ) = y + hf + h2 2 (f t + f y f) + O(h 3 ) y n+1 = y + hb 1 f + h 2 (b 1 c 1 f t + a 11 f y f) + O(h 3 ) Condition for order 1 (consistency) b 1 = 1 Conditions for order 2 c 1 = a 11 = 1/2 Conclusion Implicit Euler is of order 1 and implicit midpoint method is the only 1-stage IRK of order 2 Numerical Methods for Differential Equations p. 39/63

45 8. The stability function Applying an IRK to the test equation, we get hy i = hλ (y n + s j=1 a i,j hy j ), i = 1,...,s Let hy = [hy 1 hy s ]T and 1 = [1 1 1] T R s, then (I hλa)hy = hλ1y n so hy = hλ(i hλa) 1 1y n y n+1 = y n + s j=1 b j hy j = [1 + hλb T (I hλa) 1 1]y n Numerical Methods for Differential Equations p. 40/63

46 The stability function Theorem For every Runge-Kutta method applied to the linear test equation y = λy we have y n+1 = R(hλ)y n where the rational function R(z) = 1 + zb T (I za) 1 1. If the method is explicit, then R(z) is a polynomial of degree s The function R(z) is called the method s stability function Numerical Methods for Differential Equations p. 41/63

47 A-stability of RK methods Definition The method s stability region is the set D = {z C : R(z) 1} Theorem If R(z) maps all of C into the unit circle, then the method is A-stable Corollary No explicit RK method is A-stable (For ERK R(z) is a polynomial, and P(z) as z ) Numerical Methods for Differential Equations p. 42/63

48 A-stable methods and the Maximum Principle Theorem R(z) 1 for all z C if and only if all the poles of R have positive real parts and R(iω) 1 for all ω R This is the Maximum Principle in complex analysis Example 0 1/4 1/4 Y 1 = f(y n + hy 1 /4 hy 2 /4) 2/3 1/4 5/12 Y 2 = f(y n + hy 1 /4 + 5hY 2 /12) 1/4 3/4 y n+1 = y n + h(y 1 + 3Y 2 )/4 Numerical Methods for Differential Equations p. 43/63

49 Example... Applied to the test equation, we get y n+1 = hλ hλ (hλ)2 y n with poles 2 ± i 2 C +, and R(iω) 2 = ω ω ω4 1 Conclusion R(z) 1 z C. The method is A-stable Numerical Methods for Differential Equations p. 44/63

50 9. Linear Multistep Methods A multistep method is a method of the type y n+1 = Φ(f,h,y 0,y 1,...,y n ) using values from several previous steps Explicit Euler y n+1 = y n + hf(t n,y n ) Trapezoidal rule y n+1 = y n + h( f(t n,y n )+f(t n+1,y n+1 ) 2 ) Implicit Euler y n+1 = y n + hf(t n+1,y n+1 ) are all one-step methods, but also LM methods Numerical Methods for Differential Equations p. 45/63

51 Multistep methods and difference equations A k-step multistep method replaces the ODE y = f(t,y) by a (finite) difference equation k a k j y n j = h j=0 k b k j f(t n j,y n j ) j=0 Generating (characteristic) polynomials k ρ(w) = a m w m σ(w) = m=0 k m=0 b m w m Normalization a k = 1 or σ(1) = 1 (choose your preference) If b k = 0, the method is explicit ; if b k 0, it is implicit EE: ρ(w) = w 1, σ(w) = 1; IE: ρ(w) = w 1, σ(w) = w Numerical Methods for Differential Equations p. 46/63

52 Lagrange interpolation Given a grid {t 0,t 1,...,t k } construct a degree k polynomial basis Φ i (t) i = 0,1,...,k such that Φ i (t j ) = δ ij (the Kronecker delta) If the values z j = z(t j ) are known for the function z(t), then P(t) = k Φ j (t)z j j=0 interpolates z(t) on the grid: P(t j ) = z j ; P(t) z(t) for all t Numerical Methods for Differential Equations p. 47/63

53 Adams methods (J.C. Adams, 1880s) Suppose we have the first n + k approximations y m = y(t m ), m = 0, 1,...,n + k 1 Rewrite y = f(t,y) by integration y(t n+k ) y(t n+k 1 ) = tn+k t n+k 1 f(τ,y(τ)) dτ Adams methods approximate the integrand by an interpolation polynomial on t n,t n 1,... f(τ,y(τ)) P(τ) Numerical Methods for Differential Equations p. 48/63

54 Adams Bashforth methods (explicit) Interpolate f values with polynomial of degree k 1: P(t n+j ) = f(t n+j,y(t n+j )) j = 0,...,k 1 Then P(τ) = f(τ,y(τ)) + O(h k ) for t [t n,t n+k ] y(t n+k ) = y(t n+k 1 ) + tn+k t n+k 1 P(τ)dτ + O(h k+1 ) The k-step Adams-Bashforth method is the order k method y n+k = y n+k 1 + h k 1 j=0 b j f(t n+j,y n+j ) where b j = h 1 t n+k t n+k 1 Φ j (τ)dτ Numerical Methods for Differential Equations p. 49/63

55 Coefficients of AB1 AB1: for k = 1 y n+1 = y n + hb 0 f(t n,y n ) where b 0 = h 1 tn+1 t n Φ 0 (τ)dτ = h 1 tn+1 t n 1dτ = 1 y n+1 = y n + hf(t n,y n ) Conclusion AB1 is the explicit Euler method Numerical Methods for Differential Equations p. 50/63

56 Coefficients of AB2 Here Φ 1 (t) = t t n t n+1 t n ; Φ 0 (t) = t t n+1 t n t n+1 AB2: for k = 2 y n+2 = y n+1 + h [b 1 f(t n+1,y n+1 ) + b 0 f(t n,y n )] b 0 b 1 y n+2 = h 1 tn+2 t n+1 Φ 0 (τ)dτ = 1/2 tn+2 = h 1 Φ 1 (τ)dτ = 3/2 t n+1 [ 3 = y n+1 + h 2 f(t n+1,y n+1 ) 1 ] 2 f(t n,y n ) Numerical Methods for Differential Equations p. 51/63

57 Initializing an Adams method The first step of AB2 is y 2 = y 1 + h [ 3 2 f(t 1,y 1 ) 1 ] 2 f(t 0,y 0 ) so we need the values of y 0 and y 1 to start. y 0 is obtained from the initial value, but y 1 has to be computed with a one-step method. Implementing AB2: we may use AB1 for the first step, y n+2 y 1 = y 0 + hf(t 0,y 0 ) [ 3 = y n+1 + h 2 f(t n+1,y n+1 ) 1 ] 2 f(t n,y n ), n 0 Numerical Methods for Differential Equations p. 52/63

58 Example: AB1 vs AB2 Approximate the solution to y = y 2 cos t, y(0) = 1/2, t [0,8π] with h = π/6 and h = π/60 AB1: Solutions Errors AB2: Solutions Errors Numerical Methods for Differential Equations p. 53/63

59 How do we check the order of a multistep method? A multistep method is of order of consistency p if the local error is k a j y(t n+j ) h j=0 k b j y (t n+j ) = O(h p+1 ) j=0 Try if the formula holds exactly for polynomials y(t) = 1, t, t 2, t 3,... Insert y = t m and y = mt m 1 into the formula, take t n+j = jh: k a j (jh) m h j=0 k k b j m(jh) m 1 = h m (a j j m b j mj m 1 ) j=0 j=0 Numerical Methods for Differential Equations p. 54/63

60 Order conditions for multistep methods Theorem A k-step method is of consistency order p if and only if it satisfies the following conditions: k j m a j = m j=0 k j m 1 b j, m = 0,1,...,p j=0 k j p+1 a j (p + 1) j=0 k j p b j j=0 Then the multistep formula holds exactly for all polynomials of degree p or less. (Problems with y = P(t) are solved exactly) Numerical Methods for Differential Equations p. 55/63

61 The root condition Definition A polynomial ρ satisfies the root condition if all of its zeros have moduli less than or equal to one and the zeros of unit modulus are simple Examples ρ(w) = (w 1)(w 0.5)(w + 0.9) ρ(w) = (w 1)(w + 1) ρ(w) = (w 1) 2 (w 0.5) ρ(w) = (w 1)(w 2)(w + 2) ρ(w) = (w 1)(w ) All Adams methods have ρ(w) = w k 1 (w 1) Numerical Methods for Differential Equations p. 56/63

62 The Dahlquist equivalence theorem Definiton A method is zero-stable if ρ satisfies the root condition Theorem A multistep method is convergent if and only if it is zero-stable and consistent of order p 1 (without proof) Example k-step Adams-Bashforth methods are of order k and have ρ(w) = w k 1 (w 1) they are convergent Numerical Methods for Differential Equations p. 57/63

63 Dahlquist s first barrier Theorem The maximal order of a zero-stable k-step method is k for explicit methods p = k + 1 if k is odd for implicit methods k + 2 if k is even Numerical Methods for Differential Equations p. 58/63

64 Backward differentiation formula of order 2 Construct a 2-step method of order 2 of the form α 2 y n+2 + α 1 y n+1 + α 0 y n = h f(t n+2,y n+2 ) Order conditions for p = 2: α 2 + α 1 + α 0 = 0; 2α 2 + α 1 = 1; 4α 2 + α 1 = 4 α 2 = 3/2; α 1 = 2; α 0 = 1/2; ρ(w) = 3 2 (w 1)(w 1 ) BDF2 is convergent 3 Numerical Methods for Differential Equations p. 59/63

65 Backward differentiation formulas The backward difference operator is defined as 0 y n+k = y n+k and j y n+k = j 1 y n+k j 1 y n+k 1, j 1 Theorem (without proof): The k-step BDF method k j=1 j j y n+k = hf(t n+k,y n+k ) is convergent of order p = k if and only if 1 k 6 Note BDF methods are suitable for stiff problems Numerical Methods for Differential Equations p. 60/63

66 BDF1-6 stability regions The methods are stable outside the indicated area 20 Stability regions of BDF1 6 methods Numerical Methods for Differential Equations p. 61/63

67 A-stability of multistep methods Applying a multistep method to the linear test equation y = λy produces a difference equation k j=0 a j y n+j = hλ k j=0 b j y n+j The characteristic equation (with z := hλ) ρ(w) zσ(w) = 0 has k roots w j (z). The method is A-stable iff Rez 0 w j (z) 1, with simple unit modulus roots (root condition) Numerical Methods for Differential Equations p. 62/63

68 Dahlquist s second barrier Theorem (without proof): The highest order of an A-stable multistep method is p = 2. Of all 2nd order A-stable multistep methods, the trapezoidal rule has the smallest error Note There is no such order restriction for Runge Kutta methods, which can be A-stable for arbitrarily high orders A multistep method can be useful although it isn t A-stable Numerical Methods for Differential Equations p. 63/63

Numerical Methods for Differential Equations

Numerical Methods for Differential Equations Chapter 2: Runge Kutta and Multistep Methods Gustaf Söderlind Numerical Analysis, Lund University Contents V4.16 1. Runge Kutta methods 2. Embedded RK methods