Initial value problems method

Modelling of steady-state free surface flow corresponds to the solution of a boundary value problem while moving boundary tracking is, in general, viewed as an initial value problem. Therefore, classification of existing methods on the basis of their suitability for boundary value or initial value problems has also been advocated. 

Ritchmyer, R., and K. Morton. Difference Methods for Initial-Value Problems. 2d ed.. Interscience, New York (1967). 

Shooting Methods The first method is one that utihzes the techniques for initial value problems but allows for an iterative calculation to satisfy all the boundaiy conditions. Consider the nonlinear boundaiy value problem... 

Parabolic Equations m One Dimension By combining the techniques apphed to initial value problems and boundary value problems it is possible to easily solve parabolic equations in one dimension. The method is often called the method of lines. It is illustrated here using the finite difference method, but the Galerldn finite element method and the orthogonal collocation method can also be combined with initial value methods in similar ways. The analysis is done by example. 

Ordinaiy differential Eqs. (13-149) to (13-151) for rates of change of hquid-phase mole fractious are uouhuear because the coefficients of Xi j change with time. Therefore, numerical methods of integration with respect to time must be enmloyed. Furthermore, the equations may be difficult to integrate rapidly and accurately because they may constitute a so-called stiff system as considered by Gear Numerical Initial Value Problems in Ordinaiy Differential Equations, Prentice Hall, Englewood Cliffs, N.J., 1971). The choice of time... 

C (0). The analytieal solution to Equation 9-34 is rather eomplex for reaetion order n > 1, the (-r ) term is usually non-linear. Using numerieal methods, Equation 9-34 ean be treated as an initial value problem. Choose a value for = C (0) and integrate Equation 9-34. If C (A.) aehieves a steady state value, the eorreet value for C (0) was guessed. Onee Equation 9-34 has been solved subjeet to the appropriate boundary eonditions, the eonversion may be ealeulated from Caouc = Ca(0). 

Richtmyer, R. D. and K. W. Morton. 1967. Difference methods for initial value problems. New York Interscience. 

Shooting methods attempt to convert a boundary value problem into an initial value problem. For example, given the preceding example restated as an initial value problem for which... 

They convert the initial value problem into a two-point boundary value problem in the axial direction. Applying the method of lines gives a set of ODEs that can be solved using the reverse shooting method developed in Section 9.5. See also Appendix 8.3. However, axial dispersion is usually negligible compared with radial dispersion in packed-bed reactors. Perhaps more to the point, uncertainties in the value for will usually overwhelm any possible contribution of D. ... 

The forward shooting method seems straightforward but is troublesome to use. What we have done is to convert a two-point boundary value problem into an easier-to-solve initial value problem. Unfortunately, the conversion gives a numerical computation that is ill-conditioned. Extreme precision is needed at the inlet of the tube to get reasonable accuracy at the outlet. The phenomenon is akin to problems that arise in the numerical inversion of matrices and Laplace transforms. 

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)]. 

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. 

For the second step one establishes a solution method. The system under consideration may be static, dynamic, or both. Static cases require solving a boundary value problem, whereas dynamic cases involve an initial value problem. For the illustrative problem, we discuss the solution of a static Laplace (no sources) or Poisson (sources) equation such as... 

With a guessed value for the unknown initial condition g3(0), the initial-value problem is solved for the interval 0 < z < Z, where Z is sufficiently large as to be outside the boundary layer. For an arbitrary guess at g3(0), the value of g2(Z) — 1 will not be satisfied. An iteration procedure (e.g., a Newton method) is implemented to find the initial value for g3 for which the outer boundary value for g2 is satisfied. 

Thus the equations that we must solve are 12.196 and 12.197, which comprise a set of two coupled first-order differential equations, subject to the boundary conditions, Xj = 0.01395, and X2 = 0.00712 at z = 0 and Xj = X2 = 0 at z = Z, with the unknown fluxes Ni, N2 that must be found. This equation set could easily be solved as a two-point boundary-value problem using the spreadsheet-based iteration scheme discussed in Appendix D. However, for illustration purposes we choose to solve the equation set with a shooting method, mentioned in Section 6.3.4. We can solve the problem as an ordinary differential equation (ODE) initial-value problem, and iteratively vary Ni,N2 until the computed mole fractions X, X2 are both zero at z = Z. 

The objective of this problem is to explore the performance of stiff and nonstiff user-oriented initial-value-problem software. Acquire the Fortran source code and the documentation for Vode from www.netlib.org. The VODE package enables the user to select either stiff or nonstiff methods. 

One of the simplest algorithms to solve a boundary value problem is the "shooting method." In this method, we assume initial values needed to make a boundary value problem into an initial value problem. We repeat this process until the solution of the initial value problem satisfies the boundary conditions. Therefore, proper initial conditions for the solution of the preceding problem are... 

Approximate solutions to the initial value problem for Equations (6.17)-(6.29) are realized by means of the quasi-linearization method (Nitu et al., 2000b). 

These Runge-Kutta methods do not require information from the past, and are very versatile if the time steps need to be adjusted as the solution evolves. The stability of the RK2 is similar to the APC2, while the RK4 has less strong conditions for stability [10]. Both are ideal for initial value problems in time or in space,... 

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,... 

Instead of purely initial conditions, a mixture of initial and final conditions may be used [77—79]. However, such boundary value problems can, at least theoretically, be tranformed into initial value problems by having recourse, for example, to a shooting method (see Sect. 4.5). 

Cash JR, Karp AH (1990) A Variable Order Runge-Kutta Method for Initial Value Problems with Rapidly Varying Right-Hand Sides Transactions on Mathematical Software 16, 3 pp 201-222... 

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