The next notion is the order (Konsistenzordnung) of a method: τ(h) = O(h p ).

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1 The next notion is the order (Konsistenzordnung) of a method: Definition. Order The integration method is of order p if τ(h) = O(h p ). After several steps, local errors are accumulated, and we need another concept to describe this process: Definition.3 Global error The global error of an integration method is e n (X) = y(x) y n where X = x n fixed, n variable. The method is convergent if lim e n(x) =. n Convergence thus means: Für feiner werdendes Gitter strebt der globale Fehler The following theorem can be shown: Theorem. Convergence of one-step methods Assume f Lipschitz-continuous with constant L from (.). Furthermore, let the one-step method be of order p, i.e., Then, the global error satisfies τ(h) = O(h p ). where h max = maxh i. i e n (X) ch p e L X x max L

2 Discussion ) Ordnung globaler Diskretisierungsfehler = Ordnung lokaler Diskretisierungsfehler. Man sagt bei ESV auch kurz: Konsistenz = Konvergenz ) Variable Schrittweiten sind zugelassen. Der Verstärkungsfaktor L (el(x x ) ) ist jedoch von den Schrittweiten unabhängig. 4) Manchmal findet man die Schreibweise e(h,x) statt e n (x) für den globalen Diskretisierungsfehler, um die Abhängigkeit von der Schrittweite zu betonen. Oft besitzt e(h, x) eine asymptotische Entwicklung in h. Einschrittverfahren sind demnach einfach zu analysieren: Man ermittelt den lokalen Fehler (durch Taylorreihenentwicklung) und bekommt aus der Konsistenz bereits globale Konvergenz. 7

3 . Runge Kutta Methods Runge Kutta methods are the most popular class of one-step methods. In each step, they evaluate the slope of y in intermediate values, the so-called stages. Example.: Method of Heun K K y y x x Discretization scheme Increment function Φ Figure 4: Idea of Heun s method y = y + h(k +K ) with stages K = f(x,y ), K = f(x,y +h K ). y = y +h Φ(x,y,h), Φ(x,y,h) = [f(x,y)+f(x+h,y +hf(x,y))] = (K +K ) General notation for one step of a Runge-Kutta method (Runge Kutta Verfahren): y = y +h(b K +b K +...+b s K s ) (.) with stage values ( ) i K i = f x +c i h,y +h a ij K j, i =,...,s. (.7) j= 8

4 The method is fully specified by the number of stages s, the coefficients b i,c i for i =,...,s and a ij for i =,...,s and j i. Example.3: Heun s method revisited Number of stages s =, coefficients b =, b =, c =, c =, a =. Example.4: Modified Euler method y = y +h K mit K = f(x,y ), K = f ( x + h,y + h K ). Thus s = and b =, b =, c =, c =, a =. Butcher Tableau c c a c s a s... a ss b b... b s i.e., c A b T Examples Heun mod. Euler Classical Runge Kutta method (Runge 895, Kutta 9) 9

5 Determining the Coefficients Criteria for the coefficients: Wie kommt man auf die Koeffizienten? Die Koeffizienten sind zunächst freie Parameter des Verfahrens. Sie werden so bestimmt, dass das Verfahren möglichst gut ist, d.h. eine möglichst hohe Ordnung erreicht. Wesentliches Hilfsmittel bei der Bestimmung sind Taylorreihenentwicklungen der exakten Lösung und der numerischen Näherung. Consider the case s = with y = y +h(b K +b K ), K = f(x +c h,y ), K = f(x +c h,y +ha K ). By Taylor expansion K = f(x,y )+f x (x,y ) c h+o(h ) K = f(x,y )+f x (x,y ) c h+f y (x,y )ha K +O(h ) = f(x,y )+f x (x,y ) c h+f y f(x,y )ha +O(h ) = y = y +h(b +b )f(x,y ) +h (b c +b c )f x (x,y ) +h b a f y f(x,y )+O(h 3 ) Compare with exact solution y(x +h) = y +hf(x,y )+ h (f x +f y f)(x,y )+O(h 3 ) Order conditions for p = b +b =, b c +b c =, b a =. (.8) Particular solutions of (.8) are Heun s method and modified Euler. Note: it is impossible to achieve p = 3 with s = stages! For higher order, we thus need more stages, and the procedure for determining the coefficients becomes tedious. Remedy: study only ODEs in

6 autonomous form. Compare y = f(x,y) and ( ) ( Y y = F(Y), Y :=, F(Y) := x f(x,y) ). A numerical method should be invariant under this transformation, i.e., should give the same results. This requirement leads to the conditions c i = i j= a ij, (.9) since in the last row of F we have x +c i h = x +h i j= a ij. Summarizing, it is sufficient to determine the coefficients a ij,b i in the autonomous case, and then the nodes c i follow from (.9). Theorem. Order conditions for p = 4 Let f C p (D) and denote by A,b,c the method coefficients. The Runge- Kutta method is of order p if (.9) holds and p = : p = : p = 3 : p = 4 : s b i = () i= s b i c i = / () and () i= s b i c i = /3, i= s b i c 3 i = /4, i= s b i a ij c j = /, i,j= s b i a ij c j = / (3) and () () i,j= s b i c i a ij c j = /8, i,j= s i,j,k= b i a ij a jk c k = /4 and () (3)

7 Constructing a Method We try now to construct a method of order p = 4 with s = 4 stages. Assuming c i = a ij, we have to determine unknown coefficients such that Theorem. holds. b,b,b 3,b 4,a,a 3,a 3,a 4,a 4,a 43 The condition b i = looks like a quadrature rule, and thus we interprete b,...,b 4 as weights of a quadrature formula. Also, the c i can be viewed as nodes in the interval [,]. From bi c i = /, bi c i = /3, bi c 3 i = /4, we get that this quadrature formula is exact for all polynomials in P 3. We select Newton s 3/8-rule where c = (, /3, /3, ) T, b = (/8, 3/8, 3/8, /8) T. and compute from b i a ij c j = /, b i c i a ij c j = /8 and b i a ij a jk c k = /4 the equations a 3 =, a 4 c +a 43 c 3 = /3, b 4 a 43 a 3 c = /4. Consequently, a 43 = and a 4 =. From the definition of c i we also get a = /3,a 3 = /3,a 4 =. The result is Kutta s 3/8-rule

8 Table on the relation between number of stages and maximum order: Number of stages s max. order p Error Estimation and Stepsize Control So far, the grid x < x <... < x n < x n has been kept fixed, usually with constant stepsize h i = x i+ x i. Disadvantages: Goals for an adaptive algorithm: (i) eine vorgegebene Genauigkeit der numerischen Lösung erreicht wird, (ii) der dazu notwendige Rechenaufwand möglichst minimal wird. 3

9 Gesamtfehler p e (x)< O(h ) n h Rundungsfehler Figure 5: Global error and influence of round-off y kleine Schrittweiten x große Schrittweiten Figure : Grid adapted to solution behavior Objective of adaptive stepsize control: Choose h i such that the next discrete solution point y i+ satisfies a given tolerance criterion. In practice, this means that we only control the local error and not the global error! I.e., we compute h i such that with ǫ given. u(x i+ ) u(xi )=y i y i+ ǫ (.) Since the exact solution u(x) for the initial value u(x i ) = y i in (.) is unknown, it is replaced by an approximation ŷ i+. For this reason, we speak of error estimation (Fehlerschätzung). 4

10 In case of Runge Kutta methods, the following approach is widespread: Combine a method of order p (for y i+ ) with a method of order p+ (for ŷ i+ ). We call ŷ i+ the embedded method (eingebettetes Verfahren). Butcher table c A b T ˆbT y = y +h s b i K i i= ŷ = y +h ŝ ˆbi K i i= This idea goes back to Fehlberg, and often these schemes with embedded error estimator are called Runge Kutta Fehlberg methods. Below we will study the computation of ŷ i+ with order p + in more detail. For the moment, we assume thatŷ i+ isavailableand derive next a generaladaptive time-stepping algorithm. Given y i+ and ŷ i+, we compute the optimal stepsize h opt. in the following way: It holds The desired stepsize h opt. satisfies ŷ i+ y i+. = c h p+ (y i+ order p). c h p+ opt. = ǫ. By division we obtain and thus h p+ opt. = h p+ ǫ ŷ i+ y i+ ǫ h opt. = h p+ ŷ i+ y i+. (.) 5

11 The relation (.) between actual and desired stepsize is the basis of all variable stepsize algorithms. Note that it only makes sense as long as h is small enough. In practice, additional heuristics is required to make the control algorithm efficient and robust. In the algorithm below, we denote the method by Φ (order p) and by ˆΦ (order p+). Algorithm. Adaptive one-step method Start: x, y, h, ǫ, x end given; i = ; while x i < x end x = x i +h i ; y = y i +h i Φ(x i,y i,h i ); ŷ = y i +h iˆφ(xi,y i,h i ); e = ŷ y ; h opt. = h p+ ǫ e ρ; % safety factor ρ h opt. = min(α h i, max(βh i, h opt. )); % bounds α,β if e ǫ x i+ = x; y i+ = y; h i+ = min(h opt.,x end x i+ ); i = i+; else h i = h opt. ; % repeat step end end

12 Remarks: ) Theconstantsρ,α,β dependonthemethod. Forhigherordermethods, the bounds for the change in h may be less restrictive. ) A good norm for computing the error is ERR = n ( ) ŷ j y j, WT = ŷ n ATOL+RTOL WT j j= with absolute and relative tolerances ATOL, RTOL. Instead of e ǫ we then compare the error with, ERR. 3) In the algorithm, we proceed with y i+ as new solution. However, ŷ i+ is more accurate, and thus it has become common sense to proceed with ŷ i+. In this case, we actually do not compute an error estimator but an error indicator. Embedded Methods As example, we construct an embedded method for the classical RK-method with p = 4. The Butcher table reads then ˆb ˆb ˆb3... ˆbŝ First approach: We assume ŝ = 4 and try to determine the 4 coefficients ˆb,...,ˆb 4 such that ŷ = y +h(ˆb K +...+ˆb 4 K 4 ) 7

13 is of order 3. The order conditions from Theorem. yield ˆb +ˆb +ˆb 3 +ˆb 4 =, ˆb /+ˆb 3 /+ˆb 4 = /, ˆb /4+ˆb 3 /4+ˆb 4 = /3, ˆb3 /4+ˆb 4 / = /, with a ij and c i inserted from above. There is only one solutionˆb = b, which means that with ŝ = 4, it is impossible to construct an embedded method! Second approach: The choice ŝ = 5 gives more degrees of freedom with respect to the coefficients. However, we don t want to spend extra computational effort for evaluating another stage K 5. Fehlberg s trick: Determine c 5,a 5,...,a 54 such that K 5 (old step) = K (new step). (.) This scheme is called FSAL(FirstSameAsLast). For our method, this means K 5 (o.s.) = f(x +c 5 h,y +h a 5j K j ) = f(x +h,y +h b i K i ) = K (n.s.). Hence c 5 = and a 5j = b j for j = : 4. An analysis of the order conditions for p = 3 now shows: There is a family of weights ˆb leading to order 3. Example: The evaluation of ŷ y reduces to the simple formula ŷ y = (K (n.s.) K 4 (o.s.))/, which means that ŷ is never really computed. 8

14 Runge-Kutta Methods in MATLAB ode45 This code is based on an explicit Runge-Kutta (4,5) formula, the Dormand- Prince pair. In general, ode45 is the best function to apply as a first try for most problems. This method is sometimes also called DOPRI5(4). It has s = and ŝ = 7 stages and uses FSAL for the extra stage. Coefficients: ode3 This is an implementation of an explicit Runge-Kutta (,3) pair of Bogacki and Shampine. It may be more efficient than ode45 at crude tolerancesand in the presence of moderate stiffness. 9

15 Chapter 3 Multistep Methods 3 Multistep methods use data from previous steps to compute a new discrete approximation. More precisely, a k step method computes y i+. = y(xi+ ) from (x i k+,y i k+ ),(x i k+,y i k+ ),...,(x i,y i ),(x i,y i ). The data stems from old steps or from a starting procedure (Anlaufrechnung). 3. Adams and BDF Methods In this section, we introduce the most popular classes of multistep methods. Adams methods. This method class is derived by integratingy = f(x,y) from x i to x i+, y(x i+ ) = y(x i )+ x i+ x i f(x,y(x)) dx. (3.) The integrand in (3.) depends on the unknown solution y, and therefore we replace it by an interpolation polynomial p. We distinguish between two cases: 3

16 a) The polynomial p interpolates the data f i k+j = f(x i k+j,y i k+j ), j =,...,k f fi k+ p(x) f i fi Integration y i+ xi k+ xi xi x x i+ Figure 7: Idea of Adams-Bashforth methods The Lagrange formula gives k p(x) = f i k+j l i,j (x) with l i,j (x) = j= Inserting p(x) in the integral in (3.), we obtain k y i+ = y i +h i β i,j f i k+j, j= k ν= ν j β i,j := h i x x i k+ν x i k+j x i k+ν. x i+ x i l i,j (x) dx. Intheequidistantcasex i = x +i h,thecoefficientsβ i,j =: β j,j =,...,k do not depend on the index i, x i+ x i l i,j (x) dx = h = h k ν= ν j k ν= ν j x +ih+sh (x +(i k +ν)h) x +(i k +j)h (x +(i k +ν)h) ds k ν +s j ν }{{} =:β j Summarizing, the method reads ds. y i+ = y i +h(β f i k+ +β f i k β k f i ) (3.) 3

17 E.g., k = y i+ = y i +hf i, k = y i+ = y i +h ( f i + 3 f i), k = 3 y i+ = y i +h ( 5 f i f i + 3 f i). This class of multistep methods is called Adams Bashforth methods. b) The alternative approach requires that p interpolates f i k+j for j =,...,k,k+. Sincepdependsnowony i+,thisleadstoanimplicitmethod. f p(x) f f i i+ xi k+ xi x i+ x Figure 8: Idea of Adams-Moulton methods An analogous construction of the interpolation polynomial yields k+ p(x) = j k+ f i k+j li,j (x), l i,j (x) = In the equidistant case, this simplifies to ν= ν j x x i k+ν x i k+j x i k+ν. βj = k+ ν= ν j k ν +s j ν ds, j =,...,k + and the method reads y i+ = y i +h(β f i k++...+β k f i +β k f i+). (3.3) 3

18 E.g., k = y i+ = y i +h ( f i + f ) i+ trapezoidal rule, k = y i+ = y i +h ( f i + 8 f i + 5 f i+). This class of implicit multistep methods is called Adams Moulton methods. Note: k = makes sense here and results in y i+ = y i +hf i+ implicit Euler. Adams methods are usually implemented as predictor corrector method (Prädiktor Korrektor Schema): (i) y () i+ = y i +h(β f i k β k f i ) (start with explicit Adams Bashforth) (ii) y (l+) i+ = Ψ(y (l) i+ ), l =,,... Correct with fixed point iteration based on implicit Adams Moulton where Ψ(y) := hβ k f(x i+,y)+y i +h(β f i k++...+β k f i). The corrector iteration converges for h small to a fixed point y = Ψ(y) due to Ψ (y) = h β k f y (x i+,y) M <. 33

19 BDF methods. The BDF (Backward Difference Formulas) are based on differentation instead of integration, as in the case of Adams methods. Determine the polynomial q(x) that interpolates and satisfies (x i k+,y i k+ ),...,(x i,y i ),(x i+,y i+ ) q (x i+ ) = f(x i+,q(x i+ )) = f(x i+,y i+ ). From the Lagrange formula, we get k+ q(x) = y i k+j li,j (x) k+ q (x i+ ) = j= In the equidistant case, d dx l i,j (x) x=x i+ = h d k+ ds ν= ν j j= k ν +s j ν y i k+j l, i,j (x i+). s= }{{} =:α j The coefficients α,...,α k are independent of step i, and hence the k-step BDF method can be written as α y i k α k y i +α k y i+ = hf i+. (3.4) BDF are implicit and important for solving stiff ODEs. Examples: k = y i+ y i = hf(x i+,y i+ ) (implicit Euler) k = 3 y i+ y i + y i = hf i+ (BDF-) k = 3 y i+ 3y i + 3 y i 3 y i = hf i+ (BDF-3) 34

20 General notation for multistep methods. Instead of α y i k α k y i+ = h(β f i k β k f i +β k f i+ ) we write a linear k-step method in the form α y i +α y i α k y i+k = h(β f i +...+β k f i+k ) or in short k α l y i+l = h l= with method coefficients α l and β l. k β l f i+l (3.5) l= 3. Order and Convergence The local error of a multistep method is defined as defect (Defekt), compare Def... Definition 3. Local error of a multistep method Let y(x) be the solution of the IVP y = f(x,y), y(x ) = y. The local error of the multistep method (3.5) with constant stepsize h is defined as ( k ) τ(h) := k α l y(x i+l ) h β l f(x i+l,y(x i+l )). h l= The error τ(h) follows thus from inserting the exact solution into (3.5). l= We say that the multistep method has order p if τ(h) = O(h p ). To simplify the notation somewhat, we set α k := in the following. Why is this possible? 35

21 Theorem 3. Order conditions The multistep method α l y i+l = h β l f i+l has order p if k α l = and l= k α l l q = q l= k β l l q for q =,...,p. (3.) l= Example 3.: Consider the method Thus α =, α = 4, α = 5, β = 4, β = y i+ +4y i+ 5y i = h(4f i+ +f i ). p = : αl =, αl l β l = ( 5 )+(4 4)+( ) = p = : p = 3 : αl l β l l = ( 5 )+(4 8)+( 4) = αl l 3 3β l l = ( 5 )+(4 )+( 8) = Overall, we have a method of order p = 3. Order of method classes from above: k step Adams Bashforth Adams Moulton BDF p k k + k What is the maximum order that we can achieve? 3

Explicit One-Step Methods

Chapter 1 Explicit One-Step Methods Remark 11 Contents This class presents methods for the numerical solution of explicit systems of initial value problems for ordinary differential equations of first