Fourth Algebraic Order Methods

2 Explicit Methods. - 4.2.1 Fourth Algebraic Order Methods. - Simos82 has derived an explicit version of Numerov s method. This version has three extra layers of the form  

The values of parameters a and b are determined in order to satisfy the minimal phase-lag property. As a result of the above a method with phase-lag of order eight and with an interval of periodicity equal to (0,21.48) is produced. 

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Ixaru and Paternoster40 have developed conditionally P-stable fourth algebraic order methods of the form ... 

Implicit Methods. - 4.1.1 P-stable Methods. - 4.1.1.1 Fourth algebraic order methods. - Simos and Raptis64 have constructed P-stable methods with minimal phase-lag for the numerical solution of Schrodinger type equations. It is the first article in the literature in which a combination of the properties of P-stability and phase-lag has been obtained in order to construct methods for Schrodinger type equations. They have produced the following two families of methods ... 

Simos and Mousadis65 have considered the following family of fourth algebraic order methods... 

Exponentially Fitted Dissipative Numerov-type Methods. - Simos and Williams have developed an exponentially fitted fourth algebraic order method which has the form ... 

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 Table 1 we present the basic properties of the fourth algebraic order two-step methods. 

Table 1 Properties of the two-step fourth algebraic order exponentially-fitted methods. S = H2 H = sqn, q = 1,2,.... The quantities m and pare defined by (11). A.O. is the algebraic order of the method. Int. Per. is the interval of periodicity of the method. TMS = Thomas, Mitsou and Simos.18 SiWi = Simos and Williams.201 = Implicit. E — Explicit...

The above problem has been solved. Simos35,36 has combined two important properties for the construction of numerical methods for the numerical integration of the Schrodinger equation. These properties are the exponential fitting and the P-stabffity. The main difference of this approach compared with the approach of Coleman and Ixaru is that the coefficients of the obtained methods are dependent by one frequency (the frequency of the problem). More specifically Simos35 has derived a family of P-stable fourth algebraic order Numerov-type exponentially-fitted methods of the form ... 

Simos and Raptis41 have considered the following symmetric fourth algebraic order symmetric four-step method... 

More recently Simos and Williams42 considered the following fourth algebraic order symmetric two-step predictor-corrector method ... 

Simos49 has constructed the following family of symmetric two-step fourth algebraic order hybrid methods ... 

In [160] the authors obtained a new fourth algebraic order Rimge Kutta-Nystrom method with vanished phase-lag, amplification error and the first derivatives of the previous properties. More spedlically the authors consider the general form of the Runge-Kutta-Nystrom method... 

In [166] the authors inviestigated a non-linear explicit two-step method of fourth algebraic order with vanished phase-lag. More specifically they used the non-linear scheme ... 

The authors proved that the method of fourth algebraic order and they determined the parameter of the method c in order the obtained scheme to have vanished phase-lag. For the interval of periodicity it is easy to see that for the parameter of the method c with which the method have vanished phase-lag the method is P-stable. The authors show the efficiency of their new produced method via numerical experiments. 

In the same formula, h is the step size of the integration and n is the number of steps, i.e. is the approximation of the solution in the point x and Xn = XQ + n h and xq is the initial value point. Using the above methods and based on the theory of phase-lag and amplification error for the Runge-Kutta-Nystrom methods, the authors have derived two fourth algebraic order Runge-Kutta-Nystrom methods with phase-lag of order four and amplification error of order five. For both of methods the authors have obtained the stability regions. The efficiency of the produced methods is proved via numerical experiments. 

Simos has defined the free parameters in order for the method to be exact for the fimctions J, = krji krg), T, = kr.jy, krg), q = n - l(l)n + 1, where ji(kr), yi(kr) are the spherical Bessel and Neumann functions respectively. He also considered the fourth algebraic order explicit method ... 

The authors consider the following three cases of fourth algebraic order six stages symplectic methods ... 

Th. Monovasilis, Z. Kalogiratou and T. E. Simos, Families of third and fourth algebraic order trigonometrically fitted symplectic methods for the numerical integration of Hamiltonian systems. Computer Physics Communications, 2007, 177, 757 763. 

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. 

These methods integrates exactly the harmonic oscilators with frequency ft). Hence, the coefficients of the above type of methods are dependent on a parameter w = (oh, where h is the stepsize of integration. The author developed the order conditions for the above methods. He studied also the stability and phase properties of these type of methods. The above developed theory has been applied on fourth and fifth algebraic order schemes of the above form. Numerical illustrations show the efficiency of the above mentioned methods. 

If j30 = 0, the method is explicit and the computation of is straightforward. If 30 + 0, the method is implicit because an implicit algebraic equation is to be solved. Usually, two algorithms, a first one explicit and called the predictor, and a second one implicit and called the corrector, are used simultaneously. The global method is called a predictor-corrector method as, for example, the classical fourth-order Adams method, viz. 

Equations 10.106 to 10.111 constitute a set of algebraic equations and first order ordinary differential equations. The two algebraic Eqs. 10.104 and 10.105 are solved using the classical procedure described by Villadsen [65]. The set of ordinary differential equations is easy to solve with the fourth-order Runge-Kutta method. 

This section deals with the construction of optimal higher order FDTD schemes with adjustable dispersion error. Rather than implementing the ordinary approaches, based on Taylor series expansion, the modified finite-difference operators are designed via alternative procedures that enhance the wideband capabilities of the resulting numerical techniques. First, an algorithm founded on the separate optimization of spatial and temporal derivatives is developed. Additionally, a second method is derived that reliably reflects artificial lattice inaccuracies via the necessary algebraic expressions. Utilizing the same kind of differential operators as the typical fourth-order scheme, both approaches retain their reasonable computational complexity and memory requirements. Furthermore, analysis substantiates that important error compensation... 

The steady-state model consists of a set of coupled ordinary differential and algebraic equations. The simulation is obtained by integrating simultaneously the mass-balance equations for the gas and liquid phases in the axial direction of the reactor using a fourth-order Runge-Kutta method. The heat balance is used only for the simulation of the industrial reactors. The solid phase algebraic equations are solved between integration steps with the Newton-Raphson method. Physical properties and mass-transfer coefficients are also updated in every integration step. 

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