Hairer, E., Nrsett, S. P., and Wanner, G. Solving Ordinary Differential Equations I: Nonsti# Problems, vol. 8 of Springer Series in Computational Mathematics. Springer-Verlag, New York, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....f(t, y) the BDF methods take the form k Z a )Y( n )At) i=O Atf(nAt, y(nAt) where 1 k 6 is the order of accuracy and cl k) are known constants. Gen eralization to adaptive time stepping is relatively straight forward, and will introduce a dependence of cl k) on the time steps [17]. The BDF methods have desirable stability properties. In the sense of theory of numerical methods for ODE, the first and second order methods are A stable, and methods of order up to and including six are A(c) stable [17, 18] BDF(1) is the well known backward Euler method. BDF(k) is a k step ....

....straight forward, and will introduce a dependence of cl k) on the time steps [17] The BDF methods have desirable stability properties. In the sense of theory of numerical methods for ODE, the first and second order methods are A stable, and methods of order up to and including six are A(c) stable [17, 18]. BDF(1) is the well known backward Euler method. BDF(k) is a k step multi step method and the order of accuracy is determined by the number of previous solutions used to approximate the time derivative. Thus one can obtain a high order of accuracy in the temporal discretization with little extra ....

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E. HAIRER, S. P. NORSETT, AND G. WANNER, Solving Ordinary Differ- ential Equations I - Nonstiff Problems, Springer-Verlag, 1987.

....corresponding communication time, whereas the effect on the other machines is much larger and leads to a significant overall performance improvement of the execution time. Extrapolation methods Extrapolation methods are explicit solution methods for ODEs with a possibly high convergence order [9, 17]. They are widely used and are especially suited if high precision is required. In each time step, they compute # different approximations for the same time step with different step sizes # # , # # ######, which are combined to obtain an approximation solution of higher 100 200 300 400 500 0 ....

....results are obtained for # # #. The top part of Figure 8 shows the runtimes for # ##and # ##processors, the bottom part shows the runtimes for # and # ###processors. As example, we consider sparse ODE systems resulting from a discretization of reaction diffusion equations with different grids [9]. For all numbers of processors, one of the task parallel execution schemes leads to the smallest execution time and 0 0.5 1 1.5 2 2.5 in methods based on RadauIA of order 3, p=32, Brusselator implicit RK DIIRK data parallel DIIRK task parallel 18 50 98 162 242 338 0 1 2 3 4 5 ....

E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer--Verlag, Berlin, 1993.

.... predictorcorrector pair, before extending our results to the error per unit step (EPUS) case and backward differentiation formulae (BDF) We believe these new results are interesting because they address directly a problem, namely the solution of a class of non linear equation [7, 12], often considered intractable. Standard codes usually make simplifying assumptions about the previous stepsizes which may fail, resulting in step rejections and possible restarts, if these conditions are not met [11] This report (with erratum) appeared as pre print 93 47 at the IWR, ....

....solving an equation of the form k;n 1 (t n 1 ) f(t n 1 ; e y n 1 ) 11) for e y n 1 given f(t i ; e y i ) i ng. We refer to such methods as backward differentiation formulae (BDF) For stiff problems, 11) must typically be solved by Newton iteration. For an introduction to BDF methods see [6, 7, 8]. However, substituting the true values fy(t i )g for fe y i g in (11) and Y k;n 1 (t n 1 ) we merely note that the local truncation error n 1 b Y k;n 1 (t n 1 ) f(t n 1 ; e y(t n 1 ) n 1 7 may be written [1, 5] n 1 = h Delta n 1 (t n 1 ) Delta y[t n 1 ; t n ; t ....

Hairer, E., Nrsett, S. P. and Wanner, G. "Solving Ordinary Differential Equations I -- Nonstiff Problems." Springer-Verlag (1987).

....technique when the stepsize is large compared to the delay (t; y(t) The discussion in our paper is intended to establish that some unexpected possibilities have to be considered in the development of a robust algorithm. 2 Interpolation Runge Kutta methods can be adapted to DDEs as seen in [8]. There exists a choice of interpolants for Runge Kutta methods: continuous (embedded) extensions [8] hermite interpolants [12] natural continuous extensions (NCEs) 18] and highly continuous extensions [9] However, for a p th order Runge Kutta method, the (p 1) st and p th order NCEs and ....

....is intended to establish that some unexpected possibilities have to be considered in the development of a robust algorithm. 2 Interpolation Runge Kutta methods can be adapted to DDEs as seen in [8] There exists a choice of interpolants for Runge Kutta methods: continuous (embedded) extensions [8]; hermite interpolants [12] natural continuous extensions (NCEs) 18] and highly continuous extensions [9] However, for a p th order Runge Kutta method, the (p 1) st and p th order NCEs and continuous extensions coincide. For hermitian interpolants it has been shown that the minimal number of ....

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E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations 1: Nonstiff Problems, Springer Series in Computational Mathematics 8 (1980), Springer-Velag, Berlin.

.... jffi (t; v(t) jkD(ff(t; v(t) kg : Denoting m(t) ku(t) Gamma v(t)k and L = L y L z L ff and using the Dini derivative D m(t) lim inf h 0 m(t h) Gamma m(t) This is automatic for sufficiently small H. ffi denotes ffi for ffi 0 and zero otherwise. 9 described in [4], this then implies D m(t) Lm(t) L z jffi (t; v(t) jkD(ff(t; v(t) k: Expanding ffi(t; v(t) Gamma ff(t; v(t) Gamma ff(t; u(t) Gamma (ff(t; v(t) Gamma ff(t; u(t) it follows that ffi (t; v(t) can be bounded jffi (t; v(t) j X L ff m(t) where X = max ....

....where X = max t2[tn ;t n 1 ] j Gamma ff(t; u(t) j = O(H ) is the maximum amount by which u(t) overshoots the past discontinuity at t = This implies D m(t) Lm(t) L z L (6) where L is the bound (5) kD(t)k L . ffl Bounding the error. Using a variant of Gronwall s lemma [4], 6) yeilds the result ) by induction. L ff m(t) and defining 0 A(t) L L z L 0 B(t) L z L ff m we write D m(t) A(t)m(t) B(t) Noting that m(t) ku(t)k kv(t)k is by the boundedness of u(t) and v(t) also bounded, we write m(t) O(1) 0(H ....

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Hairer, E., Nrsett, S. P. and Wanner, G. "Solving ordinary differential equations I, non-stiff problems." Springer-Verlag, Berlin (1987).

....by derivative function evaluations. Namely g i;q = 1=q i = 1 1=q(q 1) i = 2 g i Gamma1;q Gamma ff i Gamma1 (n 1)g i Gamma1;q 1 i 3 where ff i (n 1) hn 1= hn 1 Gamma hn 1 Gammai ) Here, as below, hn = t n Gamma t n 1 (and hence above h = hn 1 ) See [11] page 85 and [6], page 373. t n 1 ; t n ; t n Gammak 1 ] OE k 1 (n 1) h n 1 (h n 1 h n ) Delta Delta Delta (h n 1 h n h n Gamma1 . h n Gammak 1 ) which is balanced by the h terms in the denominator: it approximates a solution derivative. The solution of (3) is in fact a polynomial root ....

Hairer, E., Nrsett, S. P. and Wanner, G. "Solving ordinary differential equations I -- nonstiff problems." Springer-Verlag (1987).

....at least . This requirement imposes certain conditions on the individual sets of coefficients. In addition, however, the coefficients of the separate methods are coupled together through certain compatibility requirements. The theory of deriving these requirements is covered extensively in [1, 6, 7]. Because of the assumption (8) the conditions for partitioning compatibility are a superset of those for splitting compatibility (in which more than one derivative of the linear term vanishes) 18450 140 40 9999 6 TOBIN A. DRISCOLL These conditions are similar in ....

E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, Berlin, 2nd edition, 1993.

....side evaluations can be computed in parallel. We use stepsize control and variable order based on iterated approximation of the solution. A code is developed and its performance is compared with codes based on iterated Runge Kutta methods of Gauss type and various Dormand and Prince pairs [15]. The accuracy of some of our methods are comparable with the PIRK10 methods of van der Houwen and Sommeijer [18] but require less processors. In addition at very stringent tolerances these new methods are competitive with RK78 pairs in a sequential implementation. 1 Introduction The invention ....

.... these debates, the results of van der Houwen and Sommeijer [18] showed that codes based on parallel iteration are comparable in accuracy and more efficient in terms of function evaluation than standard sequential codes such as DOPRI8 which implements high order formulas of Prince and Dormand [15]. Van der Houwen and Sommeijer s work also showed that it is possible to construct high order methods with a minimum number of stage evaluations. Their contribution motivates us to construct higher order methods draft3.0 y Department of Mathematics, University of Queensland, Australia z ....

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Hairer, E. & Wanner, G. (1987), Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Comp. Math., Springer-Verlag, Berlin.

....in some parameters which turn these problems into stiff problems. BRUS 0 problem is described in [11] and also appears in [6] but instead of using the boundary condition (4.2) applied in [11] we use the boundary condition (4.1) which will be described shortly. Whereas BRUS 1 problem is used in [2, 3, 7] and has initial conditions u(0; x; y) 2 0:25xy; v(0; x; y) 0:8x; t 2 [0; 1] 7 0 5 10 0 5 10 0 1 2 3 4 U(1.5,x,y) 0 5 10 0 5 10 0 1 2 3 4 V(1.5,x,y) Figure 1: Solution U(1:5; x; y) and V (1:5; x; y) to BRUS 0 A = 3:4; B = 1; ff = 0:002 and the boundary condition ....

....Figure 1: Solution U(1:5; x; y) and V (1:5; x; y) to BRUS 0 A = 3:4; B = 1; ff = 0:002 and the boundary condition (4.1) is applied. A solution to BRUS 0, with the boundary conditon (4. 1) and N = 15, as a function of (x i ; y i ) is given in Figure (1) while the solution to BRUS 1 can be seen in [7], pages 382 383. It is obvious that there at least two natural ways of ordering the components of the system raised from the Brusselator problem [2] These orderings, respectively (A) and (B) are u 11 ; u 1N ; u 2N ; uNN ; v 11 ; v 2N ; vNN and u 11 ; v 11 ; u ....

Hairer, E., and Wanner, G. Solving Ordinary Differential Equations I: Nonstiff Problems, springer series in comp. math. ed. Springer-Verlag, Berlin, 1987.

....( d; 3.6) since the order is p, whilst the number of terms is in general smaller than in (3.5) The number of terms in (3.6) can be further decreased by exploiting time symmetry. A flow Psi t 0 ;t is time symmetric, provided that Psi t;t 0 ffi Psi t 0 ;t = Id for every (suitably small) t t 0 [10]. It is self evident that the flow Psi t 0 ;t Y 0 = Y (t) e X(t) Y 0 , where Y is the solution of (3.1) is time symmetric. A considerably less trivial statement has been proved in [15] namely that Psi t 0 ;t Y 0 = e X [p] t) Y 0 is time symmetric [15] This has an unexpected benefit: it ....

Hairer, E., Nrsett, S.P. and Wanner, G. (1993). Solving Ordinary Differential Equations I: Nonstiff Problems (2nd rev. ed.) (Springer-Verlag, Berlin).

....results of [15] give us the general advice to choose these two algorithms in such a way that both methods are based on very similar construction principles. We thus chose an Adams Bashforth Moulton approach for both integrators. Whereas this approach is very well known for first order equations [10, 11], we shall give some more details for the fractional variant. The key to the derivation of the method is to replace the original fractional differential equation (11) by the equivalent weakly singular Volterra equation (13) and to implement a product integration method for the latter. What we do ....

....differential equations, we can speed up the algorithm somewhat. 3.4.2 Additional Corrector Iterations Recall that in the case of a very stiff equation, we mentioned that the stability properties of the Adams Bashforth Moulton integrator may not be sufficient. However, it is well known [10, 11] that the pure one step Adams Moulton method (i.e. the trapezoidal method) possesses extremely good stability properties. These are spoiled in the Adams Bashforth Moulton approach only by the fact that, in eq. 20) we cannot replace the predictor approximation y P n 1 on the right hand side by ....

Hairer E., Nrsett S. P., Wanner G. (1993) Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer, Berlin

....was extended by Hairer Lubich (1984) to a more general mechanism to expand the global error of a strictlystable multistep method. More recently, Viswanath (2000) used a similar formula to investigate global error. Moreover, the formula is reminiscent of the classical Alekseev Grobner lemma (Hairer, Nrsett Wanner 1987). A remarkable feature of the integral formula is that, once the asymptotic behaviour of a differential system and its variational equation is known, it is possible to determine 1 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 ....

.... Gamma1=4 . Higher order terms in the WKB expansion can be derived by classical techniques but they contribute little to present analysis. 5 We convert (3.1) to the vector equation y 0 = 0 1 Gammag(t) 0 y; t 0; y(0) y 0 ; 3. 3) and solve it with a p order Runge Kutta method (Hairer et al. 1987, Iserles 1996) Without loss of generality, we assume that p is even: the case of odd p lends itself to similar treatment. As is well known, the local error (y(t n ) is a linear combination, specific to the method in question, of elementary differentials of order p 1 (Hairer et al. 1987) ....

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Hairer, E., Nrsett, S. P. & Wanner, G. (1987), Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, Berlin.

....times for different sizes of the non linear equation system. Table 3 shows the result for a sparse equation system for which F has fixed evaluation costs that are independent of the system size. Such systems arize, e.g. when discretizing the spatial derivatives of the reaction diffusion equation [17]. Table 4 shows the result for a dense equation system for which the evaluation costs of F depend linearly on the system size. Such systems arize, e.g. when solving nonlinear Schrodinger equations with Galerkin Fourier methods. The tables show that the difference between the predicted and the ....

E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer--Verlag, Berlin, 1993. 22

....is determined by the location of the roots of the characteristic equation k X i=0 a i i k Gammai Gamma k X i=1 b i i k Gammai Gamma k X i=0 c i i k Gammai = 0: 2. 1) For a root i, stability requires that jij 1, with strict inequality for multiple roots, see for instance [3, 4, 7]. If this last condition is omitted, a weak, polynomial instability may occur. The requirement that jij 1 is more important, since its violation will lead to an exponential blow up. Dividing the equation by i k and making the substitution z = 1=i, the characteristic equation reads A(z) Gamma ....

E. Hairer, S.P. Nrsett, G. Wanner, Solving Ordinary Differential Equations I -- Nonstiff Problems. Springer Verlag, Berlin, 1987.

....DIIRK method. Section 3 develops different parallel implementations. Section 4 presents the numerical experiments on an Intel iPSC=860. 2 Diagonal Implicitly Iterated Runge Kutta Methods 2. 1 Runge Kutta methods Runge Kutta (RK) methods are one step solution methods for IVPs of ODEs [10] [6] [7] One step of an implicit RK method computes the next iteration vector y 1 according to the formula y 1 = y h s X l=1 b l f (v l ) 2) where the vectors v l , l = 1; s, are defined by the s Delta n dimensional fully implicit system: v l = y h s X i=1 a li f (v i ) l = ....

E. Hairer, S.P. Norsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics. Springer--Verlag, Berlin, 1993.

....t 0 ; t 1 ; t 2 ; t end with t 1 = t h : In order to achieve a good solution and to maintain a fast computation time, the stepsizes h 0 ; h 1 : have to be chosen as large as possible while guaranteeing small approximation errors. For the problem of chosing appropriate stepsizes, [5] proposes an automatic stepsize control using two different approximations y 1 and y 1 for the solution y(t 1 ) computed with the same stepsize h. The error between those two approximations error = jjy 1 Gamma y 1 jj (18) and the upper bound for the solution in the interval [t ; t 1 ....

.... when using the vectors v (j) l , l = 1; s, for one j, j m, and equation (7) y (j) y h s X l=1 b l f (v (j) l ) The solutions y (j) represent embedded solutions of successively increasing order min(p; j 1) where p is the order of the basic implicit RK method [21] [5], 12] Usually, solutions y (j) are 3 PARALLEL IMPLEMENTATION OF THE DIIRK METHOD 8 used such that the order of y 1 and y (j) differ by 1. Therefore we choose j = m Gamma 1. The error is computed according to the formula: error = jjy 1 Gamma y (m Gamma1) jj = jhj jj s X l=1 b l ....

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E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Number 8 in Springer Series in Computational Mathematics. Springer--Verlag, Berlin, 1987.

.... of the streamlines [8] continuous integration methods such as DOPRI5, which is a fifth order Runge Kutta integrator with adaptive step size monitoring and fourth order error estimation and produces a dense output directly by using informations gathered at each step of the integration [5]. At present we use the Midpoint integrator but future investigations will concern the choice of a better integration method. In particular using an adaptive step size integrator will decrease the number of integrations required, reducing overall computation time. a b c d e f Fig. 5. Image ....

E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I - Nonstiff Problems. Springer-Verlag, 1993.

....after t = 0:333 (grid refinement is limited as described in x5) so that by t = 0:4 the estimated error is about two and a half times the tolerance. Nevertheless, the computed solution has the expected profile as seen in Figure 7.6 1 . Example 3. Consider the Brusselator problem with diffusion [12] u t Gamma 1 Gamma u 2 v 4:4u = ffl Deltau; v t Gamma 3:4u u 2 v = ffl Deltav; 0 x; y; z 1; t 0; u(x; 0) 0:5 y 0:4z 0:5g(y) 0:02g(z) v(x; 0) 1:0 5x 0:25g(x) 0 x; y; z 1; ru Delta n(x; t) 0; rv Delta n(x; t) 0; x 2 Omega : 6.6) The function g( ....

E. Hairer, S.P. Norsett, and G. Wanner, Solving Ordinary Differential Equations I: Non-stiff Problems (Spring Verlag, Berlin, 1987).

....of index 1 the initial value for Z 0 is the appropriate isolated solution of the equation g(y 0 ; Z 0 (0) 0. An initial layer appears in z whenever Z 0 (0) 6= z 0 . Before we give the error bounds for integration of (2) 3) let us recall some basic concepts of linear multistep methods [17, 18, 21]. To solve a differential equation y 0 = f(t; y) by using a linear multistep formula, we take a uniform mesh of size h, t j = t 0 jh, write y j for the approximation to y(t j ) and write f j for f(t j ; y j ) The linear multistep formula is k X i=0 (ff i y n i Gamma hfi i f n i ) 0: ....

....are distinguished by having all characteristic roots equal to zero at infinity there is an additional factor =h in front of the z starting errors. Note in all cases the effect of on the errors in the fast variables z. We turn now to Runge Kutta methods and recall a few basic facts [8, 13, 17, 18, 21]. We display a Runge Kutta formula as a Butcher tableau c A b T where c and b are s dimensional vectors and A is an s Theta s matrix. The formula is used to advance the numerical solution of y 0 = f(t; y) one step by y n 1 = y n s X i=1 b i k i : 10) The scaled sample derivatives k ....

E. Hairer and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (Springer-Verlag, New York, 1987).

.... predictorcorrector pair, before extending our results to the error per unit step (EPUS) case and backward differentiation formulae (BDF) We believe these new results are interesting because they address directly a problem, namely the solution of a class of non linear equation [7, 12], often considered intractable. Standard codes usually make simplifying assumptions 1 about the previous stepsizes which may fail, resulting in step rejections and possible restarts, if these conditions are not met [11] This report (with erratum) appeared as pre print 93 47 at the IWR, ....

....an equation of the form Y 0 k;n 1 (t n 1 ) f(t n 1 ; e y n 1 ) 11) for e y n 1 given f(t i ; e y i ) i ng. We refer to such methods as backward differentiation formulae (BDF) For stiff problems, 11) must typically be solved by Newton iteration. For an introduction to BDF methods see [6, 7, 8]. However, substituting the true values fy(t i )g for fe y i g in (11) and Y 0 k;n 1 (t n 1 ) we merely note that the local truncation error n 1 b Y 0 k;n 1 (t n 1 ) f(t n 1 ; e y(t n 1 ) n 1 7 may be written [1, 5] n 1 = h Delta 0 n 1 (t n 1 ) Delta y[t n 1 ; t n ; t ....

Hairer, E., Nrsett, S. P. and Wanner, G. "Solving Ordinary Differential Equations I -- Nonstiff Problems." Springer-Verlag (1987).

....2 . Figure 8 shows the runtimes for a sparse function f which has a constant evaluation time T f = 2:2s. The lower part of the figure shows the execution time for a system of size n = 13122 (which results from a discretization of the Brusselator equation using 81 mesh points in each direction [20]) The figure shows that for some cases the difference between the measured and predicted runtime can be up to 36 . This runtime of IRK method for sparse function on SP2 IRK prediction 5000 10000 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 system size processors ....

E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer--Verlag, Berlin, 1993.

....satisfy the identity R(t2 ; t0 ) R(t2 ; t1 )R(t1 ; t0 ) and so a perturbation ffip at time t0 gets mapped to a perturbation at time t2 by the matrix matrix and matrix vector multiplication R2 ffip = R1R0 ffip. Finally, the linear map in the GHYS refinement procedure is L i = R(t i 1 ; t i ) [13]. 9 As will be seen later, it is the computation of the linear maps L i , called resolvents, that takes most of the CPU time during a refinement, because the resolvent has O(N 2 ) terms in it, and it needs to be computed to high accuracy. Presumably, if one is interested in studying simpler ....

Ernst Hairer, Syvert Paul Nrsett, and Gerhard Wanner. Solving Ordinary Differential Equations I -- Nonstiff Problems. Springer-Verlag, 1980. 59

.... the described method is p = min(p; m 1) where p is the order of the used RK method [7] The system (I) provides several approximation solution by using iteration l (j) for j m and equation (4) y (j) y h s X l=1 b l l (j) These solutions represent embedded solutions [7] [4], 5] The determination of the new stepsize uses the error = jjy 1 Gamma y m Gamma1 jj (5) and bound = max(jy j; jy 1 j) 3 Parallel Programming Model The algorithms are formulated in a coarse grain compute communicate scheme. The computations are performed according to the SPMD ....

....of Lemma 1 used with the parameters and runtime functions of the iPSC=860 result in simulations of the expected runtimes. The Figures 5, 6, 7 present the predicted runtimes of the parallel IRK algorithms 1, 2 and 3 for different numbers of processors p = 4; 8; 16. We use a 3 stage Radau method [4] of order p = 5 as corrector. Because of p = min(p; m 1) we execute 4 corrector iterations. Each of the Figures 5, 6, 7 contains the runtimes for solving a system of ODEs with two different right hand side function f : ffl (con) one that has constant evaluation time T f and ffl (lin) one ....

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E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Number 8 in Springer Series in Computational Mathematics. Springer--Verlag, Berlin, 1987.

....results of [14] give us the general advice to choose these two algorithms in such a way that both methods are based on very similar construction principles. We thus chose an Adams Bashforth Moulton approach for both integrators. Whereas this approach is very well known for first order equations [9,10], we shall give some more details for the fractional variant. The key to the derivation of the method is to replace the original fractional differential equation (11) by the equivalent weakly singular Volterra equation Fractional Differential Equations in Viscoplasticity 5 (13) and to implement a ....

Hairer E., Nrsett S. P., Wanner G. (1993) Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd edn. Springer, Berlin

....after John Butcher, provide a powerful tool for the categorisation of numerical methods for solving systems of ordinary differential equations. For example, they have been used to explore the effective order of multi derivative methods, Rosenbrock methods and compositions of Runge Kutta methods [5, 4]. Compositions of methods with their adjoints have also been studied in the symplectic case [10, 2, 7] A B series represents a numerical method in terms of graph theoretical concepts [5] Given a system of ordinary differential equations in R d dy dt = f(y) 1.1) one can express the ....

.... the effective order of multi derivative methods, Rosenbrock methods and compositions of Runge Kutta methods [5, 4] Compositions of methods with their adjoints have also been studied in the symplectic case [10, 2, 7] A B series represents a numerical method in terms of graph theoretical concepts [5]. Given a system of ordinary differential equations in R d dy dt = f(y) 1.1) one can express the Taylor expansion of the solution in the form y(t Deltat) y(t) 1 X r=1 ( Deltat) r r X ae r(ae) r ff(ae ) F(ae ) y(t) LUT Mathematical Sciences Report A 204, December ....

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E. Hairer, S. P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd Edition, Springer, Berlin, 1993.

....n h 2 s X j=1 A ij f(t n c j h; q (j) i = 1; s : 1. 3) The quantity h is the stepsize of the method which, upon a suitable choice of the parameters (c i ; A ij ; b j ; b j ) produces approximations q n q(t n ) p n dq dt (t n ) of the exact solution q(t) Following [5] the parameters characterizing such a scheme are summarized in a Butcher tableau: c 1 A 11 : A 1s . c s A s1 : A ss b 1 : b s b 1 : b s (1.4) The adjoint (reflected) map Psi h : Psi Gammah ) Gamma1 is given by the RKN method (1.2) 1.3) ....

....weights and abscissae (b j ; c j ) are given by the Gauss Legendre data, the matrix (A ij ) is determined by a linear system of simplifying equations with a 1 dimensional solution set. Finally, we list the resulting Butcher tableaus through s = 5 stages. 2 Order Theory Following Hairer et al. [5] the conditions on the parameters (c i ; A ij ; b j ) to define a method of some specific order may be formulated via special Nystrom rooted trees (sNr trees) oeae = V; E; r; colour) Here V = f1; ng is a set of labels for the vertices of the tree, E ae V Theta V is the set of edges, ....

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E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin, 1987.

....of these methods. They may be satisfied identically by imposing simple symmetries among the parameters. Further, they serve as simplifying assumptions leading to a significant reduction of the complexity of the order conditions. 2 Bi colour rooted trees and densities Following Hairer et al. [10] the conditions on the parameters of pRK and RKN schemes to define a method of some specific order may be formulated via bi colour rooted trees fiae = V; E; r; colour) Here V = f1; ng is a set of labels for the vertices of the tree, E ae V Theta V is the set of edges, r 2 V is the ....

....on the new parameters introduced by (3.6) For non separable Hamiltonian problems with a generic Hamiltonian H(q;p; t) one also needs to impose the following conditions [16, 11] b j = B j ; j = 1; s (3.10) in order to obtain a symplectic method. 3. 3 Order theory Following Hairer et al. [10] the conditions on the pRK parameters to define a method of some specific order may be formulated via pRK weights associated with bi colour rooted trees. For oeae = V; E; r; colour) these weights are given by Phi pRK (fiae ) s X n 1 =1 Delta Delta Delta s X n jV j =1 B nr i Y ....

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E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin, 1987.

....a ij have to be chosen such that y n represents an approximation of the Taylor expansion of the solution. The coefficients c i are given by the row sum conditions c i = s X j=1 a ij ; i = 1; s: 1. 3) They appear naturally in the derivation of order conditions for high order methods [10]. It has become customary to represent the free parameters of the method using a Butcher tableau [5] c 1 a 11 Delta Delta Delta a 1s . c s a s1 Delta Delta Delta a ss b 1 Delta Delta Delta b s (1.4) In the following we shall use the matrix A = a ij ....

E. Hairer, S. P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd Edition, Springer, Berlin, 1993.

....referred to as the modified global error. The above technique, where the differential equation instead of the difference equation is exploited for deriving estimates of the global error, is similar to what is often used for modeling the transportation of local errors in ODE solvers, see, e.g. [3]. Below we derive estimates of the truncation error and the global error. The convective and the diffusive parts are treated separately. Many derivations prove to be rather laborious and are facilitated by the symbolic algebra system Maple. 3.2 The truncation error for the convective part By ....

E. Hairer, S. P. Nørsett, and G. Wanner, Solving Ordinary Differential Equations I - Nonstiff Problems, Springer-Verlag, Berlin, 2nd ed., 1993.

....application specific improvements. Section 5 presents the numerical experiments on an Intel iPSC=860. 2 The Brusselator Equation The Brusselator equation is a nonlinear partial differential equation describing a chemical reaction [3] We consider the following reaction diffusion equation [2]: u t = 1 u 2 v Gamma 4:4u ff 2 u x 2 2 u y 2 (2) v t = 3:4u Gamma uv 2 ff 2 v x 2 2 v y 2 (3) 0 x 1; 0 y 1; t 0; ff = 2 10 Gamma3 ; with Neumann boundary conditions u j = 0; v j = 0; and initial conditions ....

....equation on an Intel iPSC=860. Figure 4 shows typical solutions for the resulting concentrations of the two chemical substances. In the following, we describe the results of the experiments. 5. 1 Numerical Results For the implementation on the Intel iPSC=860, we use a 3 stage Radau method [2] of order ord = 5 as corrector. Because the order of the IRK method is min(ord; m 1) we execute 4 corrector iterations. Figures 5, 6, and 7 contain tables with the measured global execution times and diagrams with the measured speedup values for p = 4, p = 8, and p = 16 processors. The given ....

E. Hairer, S.P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer--Verlag, Berlin, 1993.

....of the initial value problem x = f(x; p; t) x(t = t 0 ; p) x 0 ; 1) where f is the vector of state derivatives and x 0 is the initial state. Problem (1) typically is solved by using a numerical integration algorithm, and a large body of literature is devoted to this subject (see, e.g. [6, 13, 14, 15, 19, 20, 21]) Also, in many engineering applications, one is interested not only in the final state, but also in performance criteria computed from the trajectories x. If optimization procedures are applied in order to choose optimal design variables with respect to certain performance criteria, or if ....

E. Hairer, S. Norsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer, New York, 1987.

....the applicability of an algorithm, for example that a particular meshing algorithm is only valid for piecewise smooth 2 manifolds. And finally, reference books have been developed that precisely document the standard library algorithms in terms of the communal database of mathematical knowledge [48, 49, 61] . Indeed it is the elaborate knowledge base that characterizes the domain in which we work. The emergence of applied logic and formal methods has opened new opportunities for computers to help in more effectively employing the methodology of mathematics. We now know how to specify algorithms ....

E. Hairer, S. P. Nørsett, and G. Wanner. Solving Ordinary Differential Equations: Non-Stiff Problems, volume 1. Springer-Verlag, New York, 1987.

....step size and tmax is the maximal computation time. The required graphs are specified between graph and graphend. 4 Results Hairer has written in [3] After a bad experience with explicit Euler just before, let s try a higher order method and a more elaborate code for this example: DOPRI5 (see [2]) The numerical solution obtained for y 2 with Rtol 1 = 2 Delta 10 Gamma2 (204 steps) as well as with Rtol = 10 Gamma3 (203 steps) and Atol 2 = 10 Gamma6 . We observe that the solution y 2 rapidly reaches a quasi stationary position in the vicinity of y 0 2 = 0, which in the ....

Hairer, E., Nørset, S. P., Wanner, G.: Solving Ordinary Differential Equations I---Nonstiff Problems, Springer-Verlag, Second revised edition, 1991.

....tree t; and a(t) is a possibly vector valued coefficient associated with the tree t. and = are the unique trees of order 0 and 1, respectively, while t = t 1 ; t m ] is the tree formed by joining the root of each tree t i , i = 1; m, to a new common root for t. See [2] or [6] for a comprehensive introduction to B series and trees. 6 It is well known that the exact solution of (1) and the numerical solution (2) 3) can be written as B series: y(x 0 h) B(1; y 0 ) Y = B(OE; y 0 ) y 1 = B( y 0 ) where 1(t) 1 for all t, OE( s ; OE( A s = c; OE(t = ....

E. Hairer, S. P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Verlag, 1987.

....the Taylor series expansion of one step methods (like Runge Kutta methods) we are looking for a series (called B series) of the form Phi h (y) y ha( q )f(y) h 2 2 a( q q )f 0 (y)f(y) y X t2T h ae(t) ae(t) ff(t)a(t)F (t) y) 3. 1) Here, we use the notation of [6], Sect. II.2: T = f q ; q q ; g denotes the set of rooted trees, ae(t) the number of vertices of a tree t 2 T (also called the order of t) and the integer ff(t) counts the number of monotonic labellings of t. The functions F (t) y) are called elementary differentials of f(y) and are ....

....h (y) of (3.1) satisfies k X j=0 ff j Phi j h (y) h k X j=0 fi j f( Phi j h (y) 3.2) in the sense of formal power series. Proof. We denote the series (3.1) by B(a; y) and consider a second series of the same structure but with a(t) replaced by b(t) It is known (Theorem II.12. 6 of [6]) that the composition of these series is again a series of the same type B(b; B(a; y) B(ab; y) where (ab) t) a(t) b(t) The dots indicate expressions which only depend on trees with less than ae(t) vertices. Consequently, the jth iterate of Phi h (y 0 ) B(a; y 0 ) satisfies ....

[Article contains additional citation context not shown here]

E. Hairer, S.P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations I. Nonstiff Problems", 2nd revised edition, Springer Series in Comp. Math. 8, Springer-Verlag (1993).

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Hairer, E., Nrsett, S. P., and Wanner, G. Solving Ordinary Differential Equations I: Nonsti# Problems, vol. 8 of Springer Series in Computational Mathematics. Springer-Verlag, New York, 1991.

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E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I: Non-stiff Systems. Springer-Verlag, Berlin, 1987.

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E. Hairer, S.P. Nrsett, G. Wanner, Solving Ordinary Differential Equations I -- Nonstiff Problems. Springer Verlag, Berlin, 1987.

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E. Hairer, S. P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I: Non-stiff Systems. Springer-Verlag, Berlin, 1987.

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Hairer, E., Nrsett, S.P. and Wanner, G. (1991). Solving Ordinary Differential Equations I: Nonstiff Problems (2nd ed.). Springer-Verlag, Berlin.

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E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer-Verlag, Berlin (1987).

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E. Hairer, S.P. Nrsett and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems (2nd ed.), Springer-Verlag, Berlin (1993). 14

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E. Hairer, S.P. Nrsett and G. Wanner (1993), Solving Ordinary Differential Equations I: Nonstiff Problems (2nd ed.), Springer-Verlag, Berlin.

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E. Hairer, S.P. Nrsett, and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer--Verlag, Berlin, 1993.

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E. Hairer, S.P. Nrsett, G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd Edn, Springer Verlag, Berlin, 1993.

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E. Hairer S.P. Nrsett, and G. Wanner, Solving Ordinary Differential Equations I: Nonstiff Problems, 2nd Ed., Springer-Verlag, Berlin 1993.

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Hairer, E., and Wanner, G., Solving ordinary differential equations I-- non-stiff problems, Springer Series in Comput. Mathematics, Vol. 8, Springer-Verlag 1993.

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Hairer E., Nrsett S.P, Wanner G., Solving Ordinary Differential Equations I: Nonstiff problems, Springer series in computational mathematics, 8, Springer Verlag, 1987.

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E. Hairer, S.P. Nrsett and G. Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems. Springer-Verlag, second revised edition, 1993.

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E. Hairer, S. P. Nrsett and G. Wanner, Solving ordinary differential equations I: Nonstiff problems, Springer Verlag, Berlin, 1993.

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