Variable-Step Methods

Simos91 has derived a modification of the method of Simos.82 Here an extra layer is presented. The values of parameters a, b and c are determined in order to satisfy the minimal phase-lag property. As a result of the above a method with phase-lag of order ten and with an interval of periodicity equal to (0,31.70) is produced. Based on the above method and the method of Simos82 a variable-step procedure is developed. 

Simos74 has considered two modifications of the P-stable method of Simos.67 In the first modification an extra layer in the top of the algorithm is used. The values of parameters a, b, c are obtained in order to produce a P-stable method with minimal phase-lag. As a result of the above, all the methods of the family 

Simos92 has considered families of sixth algebraic order methods. These families are based on the formula 

Avdelas and Simos93 have considered embedded, automatically defined, families of methods. The first family of methods is a generalization of the method of Simos.82 Here, we have predictors of the form  

Simos94 has produced an embedded explicit Numerov-type fourth algebraic order method of the form  

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A except for the additional programming that is required to evaluate the agreement between calculatd and observed group concentrations (expressed by CHISQ). The METHOD statement selected a fixed step Runge-Kutta integration method rather than a variable step method because a fixed integration method is necessary for the CHISQ PROCEDURE to work properly. 

The numerical solution is accomplished with a method-of-lines approach, using a control-volume spatial discretization. The time integration can be done using Dassl, which implements an implicit, variable-order, variable-step, method based on the BDF method [46],... 

Numerical Illustrations for the Methods with Constant Coefficients and the Variable-Step Methods... 

Variable-Step Methods. - 10.2.1 Error Estimation. - It is known from the literature (see for example refs. 110 and 111 and references therein) that the local truncation error (L.T.E.) is based on the algebraic order of the method and that there are many methods for the estimation of the LTE in the integration of systems of initial-value problems. 

Table 13 RTC (real of computation (in seconds)) to calculate S 2 for the variable-step methods (l)-(20). acc = 10 6. hmax is the maximum stepsize...

New variable-step Method developed in this review 4 0.448 0.11... 

Avdelas and Simos,93 (13) the exponentially-fitted variable-step method developed by Simos,8 (14) the variable-step phase-fitted method developed by Simos,51 (15) the variable-step P-stable method developed by Simos,74 (16) the exponentially-fitted variable-step method developed by Thomas and Simos,25 (17) the variable-step Bessel and Neumann fitted method developed by Simos,43 (18) the variable-step Bessel and Neumann fitted method developed by Simos,44 (19) the new exponentially-fitted variable step method based on the new exponentially-fitted tenth algebraic order method developed in Section... 

There are twelve rate parameters which determine the transient behaviour of the CO, O2, and but-l-ene system. The parameters ki, k-i, k2, -2, 3, and Zq given in equation (6) can be determined by carrying out experiments with CO and O2 alone and then fitting the observed gas-phase transients, while kf, and k can be determined from separate experiments with pure AI2O3 support. The entire set of differential equations given in equations (6) and (11) is solved by Gear s variable order/variable step method. 

The most efficient variable-step methods for the solution of coupled differential equations arising from the Schrodinger equation is the P-stable exponentialy fitted variable-step method developed by Aguiar and Simos. Another very efficient variable-step method is the variable-step Bessel- and Neumann-fitted method of Simos. Efficient variable-step methods for the solution of the above problem are also the variable-step Bessel- and Neumann-fitted method of Simos and the variable-step exponentialy fitted method developed by Konguetsof and Simos. Finally efficient methods for the solution of the above problem are the generator and the optimized generator developed by Avdelas et... 

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