P. DEUFLHARD, Recent progress in extrapolation methods for ordinary di#erential equations, SIAM Rev., 27 (1985), pp. 505--535.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Rosenbrock solvers, as it builds up a solution from the (non autonomous) linearly implicit Euler method, i.e. y n 1 = y n (I Gamma hJ) Gamma1 hf(t n ; y n ) by Richardson extrapolation. The use of this Euler method in an extrapolation code for stiff odes was first suggested in Deuflhard [26]. A rule of thumb is that the virtue of extrapolation manifests itself most clearly when high accuracy is required (see also [19] We have included seulex in our benchmarking as the extrapolation approach is mentioned by Zlatev [27] see Section 3.4.3) as a viable one for atmospheric ode ....

P. Deuflhard. Recent progress in extrapolation methods for ordinary differential equations. SIAM Review, 27:505--535, 1985.

....: m) f parfor (l = 1; s) f Newton(F j l (f) gg ffi Update(f;j) ffi Stepsize control(x) g Figure 6: Parallel DIIRK method. 5. 4 Example 4: Extrapolation method Extrapolation methods are numerical methods for the solution of ODE systems that determine approximations of higher order [6]. This is achieved by computing several approximations j 11 ; j rr for the same point in time t k with the same basic method but with different stepsizes. The use of the Euler method EulerStep as basic method and r different stepsizes h 1 : h r leads to a solution method of ....

P. Deuflhard. Recent progress in extrapolation methods for ordinary differential equations. SIAM Review, 27:505--535, 1985.

....linear algebra, which is relatively fast in Matlab. The integration is advanced with the lower order formula, so ode23s does not do local extrapolation. To achieve the same L stability in ode15s, the maximum order would have to be restricted to 2. We have considered algorithms along the lines of [11] for recognizing when a new Jacobian is needed and we have also considered tactics like those of [40] and [42] for this purpose. This is promising and we may revisit the matter, but the current version of ode23s forms a new Jacobian at every step for several reasons. A formula of order 2 is most ....

P. DEUFLHARD, Recent progress in extrapolation methods for ordinary di#erential equations, SIAM Rev., 27 (1985), pp. 505--535.

....on QSSA in the hope of better efficiency is to consider extrapolation algorithms. Some extrapolation methods have proved to be successful for very stiff problems arising in chemistry, e.g. extrapolation based on the linearly implicit Euler method or on the linearly implicit mid point rule, see [2, 8] and [14, Section IV.9] Therefore, extrapolation cannot be a priori discarded as a viable technique for solving the stiff systems arising in atmospheric chemistry. In general for high accuracy requirements extrapolation to high order is used, but here we are mainly interested in low order ....

P. Deuflhard. Recent progress in extrapolation methods for ordinary differential equations. SIAM Review, 27:505--535, 1985.

....series. Correspondingly, some of the most widely used codes, viz. those based on Adams formulas, the backward differentiation formulas (BDFs) popularized by Gear, and extrapolation of a midpoint rule, vary the order of the method used so as to approximate the solution as efficiently as possible [35, 12]. Although the integration is advanced with the value at x j 1 , it is obvious from the derivation of Taylor series methods that they provide an approximate solution throughout the subinterval [x j ; x j 1 ] which is to say that a continuous extension is obvious. In aggregate these approximations ....

P. Deuflhard, Recent Progress in Extrapolation Methods for Ordinary Differential Equations, SIAM Review 27 (1985) 505-535.

....cannot be A stable. Those methods allowing an arbitrary number of iterations are essentially implicit. Although other, less conventional, methods exist outside these categories, the most widely used algorithms for solving stiff differential equations utilize implicitness to some extent[17, 24, 55]. This is especially so for differential algebraic systems[14] The forward Euler method applied to equation (2.50) gives the difference equation y k 1 = y k (1 Deltat) which is stable only if j1 Deltatj 1, or Deltat Gamma2= if 2 R. Thus the dominant eigenvalue of the Jacobian ....

P. Deuflhard. Recent progress in extrapolation methods for ordinary differential equations. SIAM Review, 27:505--535, 1985.

....techniques with the formerly developed software package LARKIN due to [12, 2] for the treatment of standard though large scale chemical reaction systems. Only the chemical formalism is needed as input to start the simulation. It includes sophisticated stiff ODE solvers of extrapolation type [11], pointwise evaluation of the Galerkin approximation by a fast summation algorithm, appropriate internal scaling and monitoring by truncation error norms. In order to document the efficiency of the algorithm, a comparative run of the large scale stiff integration of the original ODE system and the ....

.... ) u(t) 4.8) Here u(t ) is understood to be the approximation of the solution u(t ) For the in general nonlinear CODE u 0 s (t) f s (u 1 (t) u 2 (t) s = 1; 2; 4. 9) the adaptive Rothe method may be based on the semi implicit Euler discretization due to [11], which here reads (I Gamma A) Deltau = f(u(t) u(t ) u 1 = u Deltau (4.10) 12 with u 1 an approximation of u(t ) and A an approximate Frechet derivative of f at some not further specified argument u, which is computationally available. Formally, this approach involves solving an ....

P. Deuflhard. Recent Progress in Extrapolation Methods for Ordinary Differential Equations. SIAM Rev. 27, page 505 (1985). 14

....[E c ; g c ] is regular. In this case theorem 1 holds and the Gauss Newton method converges. It is important to use adaptive methods for the computation of the flow x Phi t (x) and the Wronskians D x Phi t (x) which are needed for the evaluation of F rsp. F 0 ; we use extrapolation codes [4]. We have realized a global inexact Gauss Newton method (see [6] and in the local case, when we have good guess values x and T at hand, we can choose between ordinary, simplified and Quasi Gauss Newton method, for details see [15] The corresponding program package PERIOD is written in C. 2 ....

P. Deuflhard. Recent Progress in Extrapolation Methods for Ordinary Differential Equations. SIAM Rev., 27:505--535, 1985.

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P. DEUFLHARD, Recent progress in extrapolation methods for ordinary di#erential equations, SIAM Rev., 27 (1985), pp. 505--535.

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P. Deuflhard. Recent progress in extrapolation methods for ordinary differential equations. SIAM Review, 27:505--535, 1985.

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