Algebra order

The flame structure that has been calculated here is illustrated schematically in Figure 5.4. There are two zones through which the temperature varies a thicker upstream zone in which the reaction rate is negligible to all algebraic orders in followed by a thinner reaction zone in which convection is negligible to the lowest order in The reaction occurs in the thin downstream zone because the temperature is too low for it to occur appreciably upstream where the reaction is occurring, the reactant concentration has been depleted by diffusion so that it is of order times the initial reactant concentration. In the formulation based on equation (42), where c(t) is sought, the convective-diffusive zone need not be mentioned... 

In ref 146 the authors present a non-standard (nonlinear) two-step explicit P-stable method of fourth algebraic order and 12th phase-lag order for solving second-order linear periodic initial value problems of ordinary differential equations. The proposed method can be extended to be vector-applicable for multi-dimensional problem based on a special vector arithmetic with respect to an analytic function. 

In ref 154 the author presents an implicit hybrid two step method for the solution of second order initial value problem. The cost of the new obtained method is only six function evaluations per step and the algebraic order is eighth. The author studies the P-stability property and the conclusion is the new method satisfies this property requiring one stage less. 

In ref. 166 a kind of trigonometrically fitted explicit two-step hybrid method which has algebraic order six is obtained. The new method has the following characteristics ... 

In ref. 172 a new embedded pair of explicit Runge-Kutta-Nystrom (RKN) methods adapted to the numerical solution of general perturbed oscillators is produced. This pair is based on the methods developed by Franco.These methods can be used for general problems. It is proved that the embedded methods have algebraic order 4 and 3. 

T. E. Simos and P. S. Williams, Exponentially-Fitted Runge-Kutta Third Algebraic Order Methods for the Numerical Solution of the SchrSdinger Equation and Related Problems, International Journal of Modern Physics C, 1999, 10(5), 839-851. 

T. E. Simos, High algebraic order methods with minimal phase-lag for accurate solution of the Schrodinger equation, Int. J. Mod. Phys. C, 1998, 9(7), 1055-1071. 

G. Psihoyios and T. E. Simos, The numerical solution of the radial Schrodinger equation via a trigonometrically fitted family of seventh algebraic order Predictor-Corrector methods, J. Math. Chem., 2006, 40(3), 269-293. 

In Section 5 trigonometrically fitted Fifth algebraic order Runge-Kutta methods are presented. For these methods we present the construction and the error analysis from which one can see that the classical method is dependent on the third power of energy, the first exponentially-fitted methods is depended on a second power of the energy and finally the second exponentially-fitted methods is depended on first power of the energy. 

Which is of fifth algebraic order and uses the fourth algebraic order formula in order to control the error. 

The behavior of the Runge-Kutta-Nystrom Symplectic method of algebraic order four developed by Sanz-Serna and Calvo12 and the behavior of the classical partitioned multistep method is similar. These methods are much more efficient that the embedded Runge-Kutta method of Dormand and Prince 5(4) (see 13). 

Definition 1 (see 13). A Runge-Kutta method has algebraic order p when the method s series expansion agrees with the Taylor series expansion in the p first terms... 

A Runge-Kutta method must satisfy a number of equations, in order to have a certain algebraic order. These equations will be shown later in this paper. 

In Table 1 we present the algebraic order equations up the algebraic order five. There are 21 unknowns totally11. In order the method to be of 4th algebraic order eight equations must be satisfied while in order the method to be of 5th algebraic order seventeen equations must be satisfied. 

So, decreasing the step-length, therefore a = wh also decreases when w remains constant, the absolute error of the 5th algebraic order decreases and tends to zero. So we have an almost fifth-order method. 

Tree theory is a convenient way to obtain a certain algebraic order equations. 

Construction of Trigonometrically-Fitted Runge-Kutta Methods. Consider the explicit Runge-Kutta method Kutta-Nystrom 14, which has fifth algebraic order and six stages. The coefficients are shown in (70). 

In 26 the authors have developed a new trigonometrically-fitted predictor-corrector (P-C) scheme based on the Adams-Bashforth-Moulton P-C methods. In particular, the predictor is based on the fifth algebraic order Adams-Bashforth scheme and the corrector on the sixth algebraic order Adams-Moulton scheme. More specifically the new developed scheme integrates exactly any linear combination of the functions ... 

The methods are called multiderivative since uses derivatives of order two, four or six. The parameters of the method are computed in order to have eighth algebraic order and minimal phase-lag. Finally, a family of eighth algebraic order multiderivative methods with phase-lag of order 12(2)18 is developed. Numerical application of the new obtained methods to the Schrodinger equation shows their efficiency compared with other similar well known methods of the literature. 

In 39 the author has developed an explicit symmetric eight-step method which is trigonometrically-fitted and is of algebraic order eight. More specifically, the... 

In 40 the authors present a new explicit Runge-Kutta method with algebraic order four, minimum error of the fifth algebraic order (the limit of the error is zero, when the step-size tends to zero), infinite order of dispersion and eighth order of dissipation i.e. they present an optimized explicit Runge-Kutta method of fourth order. The efficiency of the newly developed method is shown through the numerical illustrations of a wide range of methods when these are applied to well-known periodic orbital problems. 

In 51 the author presents a new procedure for constructing efficient embedded modified Runge-Kutta methods for the numerical solution of the Schrodinger equation. The methods of the embedded scheme have algebraic orders five and four. Applications of the new pair to the radial Schrodinger equation and to coupled Schrodinger equations show the efficiency of the approach. 

---