Numerical Solution Methods Initial Value Problems

Chapter 7 Numerical Solution Methods (Initial Value Problems)... 

Chapter 7 Numerical Solution Methods (Initial Value Problems) uniform throughout. A heat balance on the bath gives... 

This equation must be solved for yn +l. The Newton-Raphson method can be used, and if convergence is not achieved within a few iterations, the time step can be reduced and the step repeated. In actuality, the higher-order backward-difference Gear methods are used in DASSL [Ascher, U. M., and L. R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia (1998) and Brenan, K. E., S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North Holland Elsevier (1989)]. 

We tested the efficiency of our newly obtained scheme against well known methods, with excellent results. The numerical illustration showed that our method is considerably more efficient compared to well known methods used for the numerical solution of initial value problems with oscillating solutions. 

The essence of solving the problem is shown in Fig. 4. There are two ways in which the basic equations can be solved by numerical means and by analytical procedures. In general, the PDEs or ODEs that describe actual situations are nonlinear and must be solved numerically using a computer. Each PDE is transformed into a set of ODEs by the method of lines. The ODEs are reduced to the solution of initial value problems,... 

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

Absorption columns can be modeled in a plate-to-plate fashion (even if it is a packed bed) or as a packed bed. The former model is a set of nonlinear algebraic equations, and the latter model is an ordinary differential equation. Since streams enter at both ends, the differential equation is a two-point boundary value problem, and numerical methods are used (see Numerical Solution of Ordinary Differential Equations as Initial-Value Problems ). 

Dynamic simulations are also possible, and these require solving differential equations, sometimes with algebraic constraints. If some parts of the process change extremely quickly when there is a disturbance, that part of the process may be modeled in the steady state for the disturbance at any instant. Such situations are called stiff, and the methods for them are discussed in Numerical Solution of Ordinary Differential Equations as Initial-Value Problems. It must be realized, though, that a dynamic calculation can also be time-consuming and sometimes the allowable units are lumped-parameter models that are simplifications of the equations used for the steady-state analysis. Thus, as always, the assumptions need to be examined critically before accepting the computer results. 

In general, the numerical solution of PDEs is much more difficult to automate than the solution of initial-value ODEs. The best method to be used is very dependent on the problem being solved. 

The outlet concentration from a maximum mixedness reactor is found by evaluating the solution to Equation 9-34 at X = 0 CAout = CA(0). The analytical solution to Equation 9-34 is rather complex for reaction order n > 1, the (-rA) term is usually non-linear. Using numerical methods, Equation 9-34 can be treated as an initial value problem. Choose a value for CAout = CA(0) and integrate Equation 9-34. If CA(X) achieves a steady state value, the correct value for CA(0) was guessed. Once Equation 9-34 has been solved subject to the appropriate boundary conditions, the conversion may be calculated from CAout = Ca(0)-... 

In ref 156 the author studies the stability properties of a family of exponentially fitted Runge-Kutta-Nystrom methods. More specifically the author investigates the P-stability which is a very important property usually required for the numerical solution of stiff oscillatory second-order initial value problems. In this paper P-stable exponentially fitted Runge-Kutta-Nystrom methods with arbitrary high order are developed. The results of this paper are proved based on a S5unmetry argument. 

In ref 171 new Runge-Kutta-Nystrom methods for the numerical solution of periodic initial value problems are obtained. These methods are able to integrate exactly the harmonic oscillator. In this paper the analysis of the production of an embedded 5(3) RKN pair with four stages is presented. 

Yonglei Fang and Xinyuan Wu, A trigonometrically fitted explicit Numerov-type method for second-order initial value problems with oscillating solutions, Applied Numerical Mathematics, in press. 

Numerical illustrations show that the procedure of trigonometric fitting is an efficient way to produce numerical methods for the solution of second-order linear initial value problems (IVPs) with oscillating solutions. 

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