Numerical Solution of Partial Differential Equations

Numerical Solution of Partial Differential Equations Chapter 6 

For heat conduction in solids, where the velocity terms are zero, Eq. (6.6) simplifies considerably. When combined with Eq, (6.7), it gives the well-known three-dimensional unsteady-state heat conduction equation 

The equation of continuity for component A in a binary mixture (components A and B) of constant fluid density p and constant diffusion coefficient is 

10) is the three-dimensional unsteady-state diffusion equation, which has the same form as the respective heat conduction equation (6.8). 

The numerical methods for partial differential equations can be classified according to the type of equation (see Partial Differential Equations ) parabolic, elliptic, and hyperbolic. This section uses the finite difference method to illustrate the ideas, and these results can be programmed for simple problems. For more complicated problems, though, it is common to rely on computer packages. Thus, some discussion is given to the issues that arise when using computer packages. 

Parabolic Equations in One Dimension By combining the techniques applied to initial value problems and boundary value problems it 

Example Consider the diffusion equation, with boundary and initial conditions. 

We denote by c, the value of c(xk t) at any time. Thus, c, is a function of time, and differential equations in c, are ordinary differential equations. By evaluating the diffusion equation at the ith node and replacing the derivative with a finite difference equation, the following working equation is derived for each node i,i = 2.n (see Fig. 3-50). cl + 1-2c, + ci 1 

This can be written in the general form of a set of ordinary differential equations by defining the matrix AA. 

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Lapidus, L. and Pinder, G. F., 1982. Numerical Solution of Partial Differential Equations in Science and Engineering, Wiley, New York. 

Smith, G.D., 1985. Numerical Solution of Partial Differential Equations Finite Difference Methods, 3rd edition. Clarendon Press. 

Morton, K. and Mayers, D. (1994) Numerical Solution of Partial Differential Equations. An Introduction. Cambridge University Press Cambridge. 

Thompson, ). F., Warsi, Z. U., Mastin, C. W., Boundary-fitted coordinate systems for numerical solution of partial differential equations, ). Comput. Phys. 

The single-column process (Figure 1) is similar to that of Jones et al. (1). This process is useful for bulk separations. It produces a high pressure product enriched in light components. Local equilibrium models of this process have been described by Turnock and Kadlec (2), Flores Fernandez and Kenney (3), and Hill (4). Various approaches were used including direct numerical solution of partial differential equations, use of a cell model, and use of the method of characteristics. Flores Fernandez and Kenney s work was reported to employ a cell model but no details were given. Equilibrium models predict... 

Temperature profiles can be determined from the transient heat conduction equation or, in integral models, by assuming some functional form of the temperature profile a priori. With the former, numerical solution of partial differential equations is required. With the latter, the problem is reduced to a set of coupled ordinary differential equations, but numerical solution is still required. The following equations embody a simple heat transfer limited pyrolysis model for a noncharring polymer that is opaque to thermal radiation and has a density that does not depend on temperature. For simplicity, surface regression (which gives rise to convective terms) is not explicitly included. 

Finite-Difference Methods. The numerical analysis literature abounds with finite difference methods for the numerical solution of partial differential equations. While these methods have been successfully applied in the solution of two-dimensional problems in fluid mechanics and diffusion (24, 25), there is little reported experience in the solution of three-dimensional, time-dependent, nonlinear problems. Application of these techniques, then, must proceed by extending methods successfully applied in two-dimensional formulations to the more complex problem of solving (7). The various types of finite-difference methods applicable in the solution of partial differential equations and their advantages and disadvantages are discussed by von Rosenberg (26), Forsythe and Wasow (27), and Ames (2S). 

Smith, G.D. Numerical solution of partial differential equations. Finite difference methods. 3rd ed. Oxford Clarendon Press 1986... 

O Brien CG, Hyman MA, Kaplan S (1959) A study of the numerical solution of partial differential equations. J Math Phys 29 223-251... 

Smith GD (1985) Numerical Solution of Partial Differential Equations Finite Difference Methods. Third edition. Clarendon Press, Oxford Smolarkiewicz PK (1983) A simple positive definite advection scheme with small implicit diffusion. Mon Wea Rev 11 479-486... 

Dissinger, G. R. GRD1 - A New Implicit Integration Code for the Numerical Solution of Partial Differential Equations on Either Fixed or Adaptive Spatial Grids, Doctoral Dissertation, Lehigh Univ. Bethlehem, PA, 1983. 

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