Finite-difference methods iteration

Errors are proportional to At for small At. When the trapezoid rule is used with the finite difference method for solving partial differential equations, it is called the Crank-Nicolson method. The implicit methods are stable for any step size but do require the solution of a set of nonlinear equations, which must be solved iteratively. The set of equations can be solved using the successive substitution method or Newton-Raphson method. See Ref. 36 for an application to dynamic distillation problems. 

Elliptic Equations Elhptic equations can be solved with both finite difference and finite element methods. One-dimensional elhptic problems are two-point boundary value problems. Two- and three-dimensional elliptic problems are often solved with iterative methods when the finite difference method is used and direct methods when the finite element method is used. So there are two aspects to consider howthe equations are discretized to form sets of algebraic equations and howthe algebraic equations are then solved. 

Packages to solve boundary value problems are available on the Internet. On the NIST web page http //gams.nist.gov/, choose problem decision tree and then differential and integral equations and then ordinary differential equations and multipoint boundary value problems. On the Netlibweb site http //www.netlib.org/, search on boundary value problem. Any spreadsheet that has an iteration capability can be used with the finite difference method. Some packages for partial differential equations also have a capability for solving one-dimensional boundary value problems [e.g. Comsol Multiphysics (formerly FEMLAB)]. 

G.R. Hadley and R.E. Smith, Full-vector waveguide modeling using an iterative finite-difference method with transparent boundary conditions, J. Lightwave Technol. 13, 465-469 (1995). 

These equations are now in convenient form for iterative solution by finite-difference methods, since the free boundary has been immobilized. An implicit method is developed, based on central time differences. Denoting the dimensionless concentration gradient at the moving boundary by... 

Steps (ii) to (iv) are repeated until the solid temperature distribution is calculated within the given threshold of accuracy (i.e. the variation of solid temperature from one iteration to the next one is smaller then the desired threshold). During each step of the relaxation procedure, the integration of the balance equations is done through a finite difference method. 

This section will illustrate the tools taught in the above sections in the form of examples applied to steady state problems. Example 8.3 applies the finite difference method to a simple one-dimensional fin cooling problem and illustrates the nature of the system of equations that is normally achieved. Example 8.4 present a 2D compression molding problem where an iterative solution method is introduced. 

The set of Eqs. (38)-(40), together with appropriate boundary conditions, was solved numerically using a combination of a finite difference method and a Newton-Raphson iteration [47]. 

The mathematical models of the reacting polydispersed particles usually have stiff ordinary differential equations. Stiffness arises from the effect of particle sizes on the thermal transients of the particles and from the strong temperature dependence of the reactions like combustion and devolatilization. The computation time for the numerical solution using commercially available stiff ODE solvers may take excessive time for some systems. A model that uses K discrete size cuts and N gas-solid reactions will have K(N + 1) differential equations. As an alternative to the numerical solution of these equations an iterative finite difference method was developed and tested on the pyrolysis model of polydispersed coal particles in a transport reactor. The resulting 160 differential equations were solved in less than 30 seconds on a CDC Cyber 73. This is compared to more than 10 hours on the same machine using a commercially available stiff solver which is based on Gear s method. 

Methods applying reverse differences in time are called implicit. Generally these implicit methods, as e.g. the Crank-Nicholson method, show high numerical stability. On the other side, there are explicit methods, and the methods of iterative solution algorithms. Besides the strong attenuation (numeric dispersion) there is another problem with the finite differences method, and that is the oscillation. 

For brevity, further discussion is restricted to the spatial discretization used to obtain ordinary differential equations. Often the choice and parameters selection for this methods is left to the user of commercial process simulators, while the numerical (time) integrators for ODEs have default settings or sophisticated automatic parameter adjustment routines. For example, using finite difference methods for the time domain, an adaptive selection of the time step is performed that is coupled to the iteration needed to solve the resulting nonlinear algebraic equation system. For additional information concerning numerical procedures and algorithms the reader is referred to the literature. 

Since this problem is stiff, an error tolerance of 10 was used. It has to be noted that iterative finite difference method does not predict the middle steady state predicted in example 3.2.2. 

It is possible to program the finite difference method in Excel and use the Calculation feature to handle the circular reference. Turn off the iteration, prepare the spreadsheet, and then turn the calculation back on. Whether this converges depends upon the initial guess. 

The values of Sj,Ej are calculated from Equations 13.37 and 13.38 (combined with Equation 13.4) at current values of the variables, and the derivatives are calculated by finite difference methods. Equations 13.41 and 13.42 are solved for AT] and Al] by inverting the Jacobian matrix, the matrix of partial derivatives of Sj and Ej with respect to Tj and 1]. The independent variables Tj and 1] are then updated for the next iteration ... 

To numerically solve equations of the above mathematical models, the general computational gas dynamics is adopted in the present work. The general differential equations (2.7) and (2.31) are then discretized by the control volume-based finite difference method, and the resulting set of algebraic equations is iteratively solved. The numerical solver for the general differential equations can be repeatedly appUed for each scale variable over a controlled volume mesh. This process must be conducted extremely carefully to avoid the influence of slight changes in the accuracy of discretization. 

Finite difference methods are the traditional approach to the numerical solution of partial differential equations. The spatial regime is divided into a grid, as shown in Figure 1. Partial derivatives are approximated by differences the derivative duldx at grid point ij) might be approximated as [u j - ut ijVh, for example. The partial differential equations are thus reduced to a set of algebraic equations for the values of the dependent variables at the grid points. Iterative... 

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