Higher Order Integration Methods

Large-order chemical engineering problems, when solved numerically, usually give rise to a set of coupled differential-algebraic equations (DAE). This type of coupling is more difficidt to deal with compared to coupled algebraic equations or coupled ODEs. The interested reader should refer to Brenan et al. (1989) for the exposition of methods for solving DAE. 

Bailey, H. E., Numerical Integration of the Equation Governing the One-Dimensional Flow of a Chemically Reactive Gas, Phys. Fluid 12, 2292-2300 (1%9). 

Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial Value Problems in Differential-Algebraic Equations, Elsevier, New York (1989). 

Burden, R. L., and J. D. Faires, Numerical Analysis, PWS Publishers, Boston (1981). 

Caillaud, J.B., and L. Padmanabhan, An Improved Semi-Implicit Runge-Kutta Method for Stiff Systems, Chem. Eng. J. 2, 22 -2i2 (1971). 

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We now use this formalism to demonstrate the derivation of a higher order integration method than the explicit Euler one. In the explicit Euler method we neglect the time-variation of a and b over the time step. This is particularly bad for the second integral as dWi is of order and thus the explicit Euler method is only 1/2-order accurate for predicting the actual trajectory. Thus, let us increase die order of accuracy of this term by using a t accurate expansion ofbmtk[Pg.345]

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

There are several methods that we can use to increase the order of approximation of the integral in eqn. (8.72). Two of the most common higher order explicit methods are the Adams-Bashforth (AB2) and the Runge-Kutta of second and fourth order. The Adams-Bashforth is a second order method that uses a combination of the past value of the function, as in the explicit method depicted in Fig. 8.19, and an average of the past two values, similar to the Crank-Nicholson method depicted in Fig. 8.21, and written as... 

S.-T. Chun and J. Y. Choe, A higher order FDTD method in integral formulation, IEEE Trans. Antennas Propag., vol. 53, no. 7, pp. 2237—2246, July 2005. 

A higher-order, nonsymplectic method used to integrate Newton s equations of motion is the Gauss-Radau method. " Similarly to the Runge-Kutta method, this technique divides the time step. At, into substeps, h, and, by using the forces that are evaluated at each h, provides accurate integration of the positions and momenta. The mass weighted force (i.e., acceleration) over a time step is expanded in the substeps, h, as... 

Therefore only two-electron integrals, as in the case of the Cl method, and triangle integrals have to be computed. This fact will be extremely helpful when extending the application of Hy-CI method to larger systems. In our code, the same computer memory is needed for Cl and Hy-CI calculations. Note, that in the Hylleraas-CI method for any electron number no higher order integrals than four-electron ones appear. 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

The price we pay for more accurate higher order methods such as the classical Runge-Kutta method has to be paid with the effort involved in their increased number of function evaluations. Many different Runge-Kutta type integration formulas exist in the literature for up to and including order 8, see the Resources appendix. 

Time-integration schemes other than BDF require other expressions of dC/dT to be consistent with the time integration scheme itself. For CN, this cannot be done consistently very well. Bieniasz showed how to do it for extrapolation and for the Rosenbrock ROWDA3 scheme [108]. The reader is referred to that paper for details, where still higher-order forms are found. The paper makes it clear that extremely small errors can be achieved by using this method. 

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