Shampine, L.F., Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, NY, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....within a given range of accuracy, different tolerances should be assign to each initial conditions. A global control of the integration would be of great help. Unfortunately, it is well known that such a control is not in general possible due to the instability factors of the ODE ( SW76] and [Sha94], pp 76 78 for a comment on Airy s equation) and the limited arithmetic precision of computers [AL97] Nevertheless, using a global error estimation technique together with an interactive visualization tool of the families of solutions, it is possible to assess the whole flow to stay within the ....

L. Shampine. Numerical Solution of Ordinary Differential Equations. Chapman and Hall, 1994.

....is slow (stiff systems) or there is a mass matrix. is a constant mass matrix. ode23t Moderately Stiff Low If the problem is only moderately stiff and you need a solution without numerical damping. is a mass matrix. The algorithms used in the ODE solvers vary according to order of accuracy [5] and the type of systems (stiff or nonstiff) they are designed to solve. See Algorithms for more details. It is possible to specify tspan, y0, and options in the ODE file (see odefile) If tspan or y0 is empty, then the solver calls the ODE file [tspan,y0,options] F( init ) to obtain any ....

Shampine, L. F. , Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.

....time step is, for example, equal to 14 for TOL = 10 Gamma5 . In view of the fact that we are solving a parabolic equation and gradient equations, this work load is still moderate. Of further interest is that the step size and local error control can be seen to obey the theory given in Shampine [9], p. 339. This theory says that upon reducing TOL by 10, the global error will asymptotically decrease by 10 p= p 1) For p = 2, the order of consistency of RKC, this gives a factor of about 5, which we can trace in Table 3. Maximum norm errors and integration statistics on 80 Theta 80 grid t ....

L.F. Shampine, Numerical solution of ordinary differential equations, Chapman & Hall, New York, 1994.

....size of the splitting step for the explicit method. This doubles the stability and positivity domain of the explicit method and hence is expected to lead overall to less time steps. 5 Numerical Experiments Following standard practice, we have implemented the four methods with variable step sizes [10, 19]. The embedded first order solution of the Rosenbrock scheme is used to obtain an estimate of the local error of the current step in the two AMF methods. The time step is selected on the basis of an error per step (EPS) control which aims to keep this estimate below a mixed (relative and absolute) ....

L.F. Shampine. Numerical solution of ordinary differential equations. Chapman & Hall, New York, 1994.

....error e n at time level t n , i.e. the difference between the computed solution at time level t n and the exact solution at the same time level. This error is in fact proportional to the tolerance T ol that we imposed, i.e. e n T ol: This property of tolerance proportionality follows from [12], p. 350, when we identify our scheme as an XEPS scheme, i.e. an error per step control with local extrapolation. The proportionality between the imposed tolerance and the global time error is a nice property since it allows the user to control the global error in a very direct manner. 5.3 ....

L. F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.

.... makes use of a stored history array of values for each component, consisting of y N (t j ) k j y 0 N (t j ) k 2 j 2 y 00 N (t j ) # ; 4) and the predictor corrector approach outlined below to compute the new solution and its time derivatives at the next time step (see [24]) The predicted values y P N (t j ) y 0P N (t j ) and y 00P N (t j ) are defined by: y P N (t j ) y N (t j Gamma1 ) k j y 0 N (t j Gamma1 ) 1 2 k 2 j y 00 N (t j Gamma1 ) y 0P N (t j ) y 0 N (t j Gamma1 ) k j y 00 N (t j Gamma1 ) and y 00P N (t ....

L. F. Shampine, Numerical solution of ordinary differential equations (Chapman and Hall, 1994).

....some initial configuration to a stationary point. Ideally, a numerical integration would relax the configuration to within some small neighbourhood of a stationary point in as few iterations as possible. However, ODE systems of the type being considered can exhibit a property known as stiffness [68, 69, 70]. Numerical solution of a stiff system of ODE s using a standard solution technique (of which Euler s method is the simplest example) is characterised by at best an excessively large number of iterations and at worst numerical instability leading to wild oscillations in the solution. The ....

L. F. Shampine. Numerical Solution of Ordinary Differential Equations. Chapman and Hall, 1994.

....variable stepsize algorithms. However, in order to state precise results we focus on explicit Runge Kutta (erk) embedded pairs. We describe below the main details of a typical adaptive erk algorithm of the type found in numerical software libraries. Further details can be found, for example, in [4, 10]. Let t n denote a sequence of (unequally spaced) grid points in time and let Un denote an approximation to u(t n ) Given Un and a stepsize Deltat n : t n 1 Gamma t n , the erk pair is defined by Y i = Un Deltat n i Gamma1 X j=1 a ij f(Y j ) 1 i s; 2.1) Un 1 = Un Deltat n s X ....

....order to compute Y i , i = 1; s. PS2 control (3.3) does require the evaluation of f(Un 1 ) but if the step is accepted then this value is needed on the next step. Also, erk pairs with the first sameas last property automatically compute f(Un 1 ) as a stage value for the current step [4, 10]. A third feature of conditions (3.2) and (3.3) is that, for any consistent method, the left hand sides of the inequalities are O( Deltat l n ) for some l 2, whilst the right hand sides are O( Deltat n ) Hence, away from fixed points, we would expect the conditions to hold automatically for ....

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L. F. Shampine. Numerical Solution of Ordinary Differential Equations. Chapman and Hall, 1994.

....integration will be affected by the error, without any indication of it. If it is too small, probably also the following ones will result too small (because all the algorithms allows only small changes to the stepsize) so that for a while an inefficient stepsize will be used (see section 7. 6 in [24]) Users usually don t wish to be involved in the choice of an initial stepsize, and often they are not able to provide a good value, so that an automatic selection of the initial stepsize is necessary. Some bibliography is available about this matter, to which we refer [16] 24] 25] 26] 31] in ....

....(see section 7.6 in [24] Users usually don t wish to be involved in the choice of an initial stepsize, and often they are not able to provide a good value, so that an automatic selection of the initial stepsize is necessary. Some bibliography is available about this matter, to which we refer [16][24][25] 26] 31] in which automatic selection of the initial stepsize are described. The algorithms used to compute the initial stepsize, if the user doesn t supply any, can be of two kind. In some codes, a fixed fraction of the length interval, like 50 or 1 , is used as initial stepsize. In other ....

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Shampine L.F., Numerical solution of ordinary differential equations, Chapman & Hall New York 1994.

.... (1) The solutions x(t) x 1 (t) x 2 (t) x 3 (t) T is desired for times 0 t 40 [5] This system is well known to be stiff, see for instance [3] Because of stability problems arising close to the equilibrium solution for much larger final times than 40, it has also been studied in [4] and [6] as well as in the documentation of the advanced numerical integration package VODE [1] However, in an effort to provide a more intermediate treatment of the problem, this note assumes the position that the system is only suspected to be stiff and that the solution is only desired on the original ....

....non stiff solver to the problem providing what can be viewed as a practical stiffness test. Section 5 continues by contrasting this performance to the one of a more appropriate stiff solver. Note that these analyses are required before attempting to solve the problem for larger times as in [4] and [6], hence this note attempts to bridge the gap to understanding those more advanced treatments. 2 The Chemical Model. The purpose of chemical models is to describe quantitatively the depletion and generation of the participating chemical reactants. Most often in applications, this is an ....

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L. F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York,

....is halted. Since we are only concerned with convergence results in real arithmetic we shall not consider this further. The crucial difference, as far as convergence properties are concerned, is that the routine uses a different embedded Runge Kutta pair, namely the Bogacki Shampine (2,3) pair [10, 9, 2]. This is designed to be operated in extrapolation mode and is a FSAL method (First Same As Last) so that while the higher order method has 3 stages, the lower order method uses the first stage of the next step and thus in reality has 4 stages. This means that the convergence results proved here ....

L.F. Shampine. Numerical Solution of Ordinary Differential Equations. Chapman and Hall, New York, 1994.

....When solving problems with y 2 R or y 2 C, the usual definition of stiffness (for systems) does not apply. However, it is still possible for a scalar ODE to be stiff. Consider y 0 (t) k(y(t) Gamma p(t) p 0 (t) y(0) A with solution y(t) A Gamma p(0) e kt p(t) see page 384 of [5]. If p(t) is smooth and slowly varying and k is large and negative this problem is stiff. We have solved this problem using rksuite 90 for p(t) cos(t) A = 0 and k = Gamma10 n ; n = 1; 2; 3; Stiffness is flagged for all n 2 on long enough ranges of integration. So the default ....

L.F. Shampine, "Numerical Solution of Ordinary Differential Equations", Chapman and Hall, 1994.

....[2] When solving problems with y 2 R or y 2 C, the usual definition of stiffness (for systems) does not apply. However, it is still possible for a scalar ODE to be stiff. Consider y 0 = J(y Gamma p(x) p 0 (x) y(0) A with solution y(x) A Gamma p(0) e Jx p(x) see page 384 of [5]. If p(x) is smooth and slowly varying and J is large and negative this problem is stiff. We have solved this problem using rksuite 90 for p(x) cos(x) A = 0 and J = Gamma10 n ; n = 1; 2; 3; Stiffness is flagged for all n 2 on long enough ranges of integration. So the default ....

L.F. Shampine, "Numerical Solution of Ordinary Differential Equations", Chapman and Hall, 1994.

....the SLATEC and NAG libraries and specific codes written by ourselves and others serve as mileposts. This survey is necessarily brief; basic definitions and fundamental algorithms can be found in other papers of this volume and some issues are treated in more detail but from a similar viewpoint in [43]. The history of developments that led to features found in the suite of codes RKSUITE [7, 8] provides the skeleton of this survey. It is fleshed out with specific developments in formulas, initial step size selection, the selection of step size for efficiency and control of the local error, ....

....relatively expensive computation of the dominant eigenvalues of a local Jacobian (using a nonlinear power method) periodically and then only when very inexpensive tests suggest that stiffness is possible. 6 Continuous Extension Adams formulas and the BDFs are based on interpolating polynomials [43], and this makes it clear how to approximate the solution between steps. Modern codes based on these formulas integrate with about the largest step size that will provide the specified local accuracy and keep the computation stable. They obtain output at specific points by evaluating the ....

L.F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.

....[31] We are still actively developing software with, e.g. the first author leading the development of the MATLAB ODE Suite [38] and the second involved in the ODE software used by the TI 85 [29] and TI92 calculators. In addition to developing software, recent publications include a monograph [35] on the practical numerical solution of IVPs and a paper [36] on the historical development of IVP software based on explicit Runge Kutta methods. In addition to our experience in research and development into numerical methods, we have taught courses on ODEs and on numerical methods at all ....

....a combination of the two. It would suffice just to show students how to introduce variables so as to formulate a problem as a system of first order equations, but an example makes the point that there is more than one way to prepare a problem for its numerical solution. This example is taken from [35] where other aspects of the task are considered. Suppose we wish to solve (p(x)y 0 ) 0 q(x)y = g(x) p(x) 0, an equation that arises in many physical contexts. One way to proceed is first to write the equation as y 00 p 0 (x) p(x) y 0 q(x) p(x) y = g(x) p(x) and then to ....

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L.F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.

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Shampine, L.F., Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, NY, 1994.

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L. F. Shampine. Numerical Solution of Ordinary Differential Equations, Chapman & Hall, London, 1993.

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L. F. Shampine. Numerical Solution of Ordinary Differential Equations, Chapman & Hall, London, 1993.

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L.F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, New York, 1994.

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Lawrence F. Shampine. Numerical solution of ordinary differential equations. Chapman and Hall, 1994.

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Lawrence F. Shampine. Numerical solution of ordinary differential equations. Chapman and Hall, 1994.

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Lawrence F. Shampine. Numerical solution of ordinary differential equations. Chapman & Hall, New York, 1994.

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L. F. Shampine. Numerical Solution of Ordinary Differential Equations. Chapman & Hall, New York, 1994.

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L.F. Shampine, Numerical Solution of Ordinary Differential Equations, Chapman & Hall, 1994. Some general texts on numerical methods Misecellaneous Exercises 51

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