Partial differential equations numerical methods

Ames, W. F. (1992). Numerical Methods Partial Differential Equations, 3rd Edition, Academic Press. 

See also Numerical Analysis and Approximate Methods and General References References for General and Specific Topics—Advanced Engineering Mathematics for additional references on topics in ordinary and partial differential equations. 

Discretization of the governing equations. In this step, the exact partial differential equations to be solved are replaced by approximate algebraic equations written in terms of the nodal values of the dependent variables. Among the numerous discretization methods, finite difference, finite volume, and finite element methods are the most common. Tlxe finite difference method estimates spatial derivatives in terms of the nodal values and spacing between nodes. The governing equations are then written in terms of... 

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

Although only approximate analytical solutions to this partial differential equation have been available for x(s,D,r,t), accurate numerical solutions are now possible using finite element methods first introduced by Claverie and coworkers [46] and recently generalized to permit greater efficiency and stabihty [42,43] the algorithm SEDFIT [47] employs this procedure for obtaining the sedimentation coefficient distribution. 

A marching-ahead solution to a parabolic partial differential equation is conceptually straightforward and directly analogous to the marching-ahead method we have used for solving ordinary differential equations. The difficulties associated with the numerical solution are the familiar ones of accuracy and stability. 

Ames, VV. (1977) Numerical Methods for Partial Differential Equations. Academic Press New York-London. 

Usually the finite difference method or the grid method is aimed at numerical solution of various problems in mathematical physics. Under such an approach the solution of partial differential equations amounts to solving systems of algebraic equations. 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

At this point we can summarize the steps required to implement the Gauss-Newton method for PDE models. At each iteration, given the current estimate of the parameters, ky we obtain w(t,z) and G(t,z) by solving numerically the state and sensitivity partial differential equations. Using these values we compute the model output, y(t k(i)), and the output sensitivity matrix, (5yr/5k)T for each data point i=l,...,N. Subsequently, these are used to set up matrix A and vector b. Solution of the linear equation yields Ak(jH) and hence k°M) is obtained. The bisection rule to yield an acceptable step-size at each iteration of the Gauss-Newton method should also be used. 

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. 

See Partial Differential Equations. ) If the diffusion coefficient is zero, the convective diffusion equation is hyperbolic. If D is small, the phenomenon may be essentially hyperbolic, even though the equations are parabolic. Thus the numerical methods for hyperbolic equations may be useful even for parabolic equations. 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

In general, there is no analytical solution for the partial differential equations above, and numerical methods must be used. However, we can obtain analytical solutions for the simplified case represented by the shrinking-core model, Figure 9.1(a), as shown in Section 9.1.2.3. 

Analytical solution is possible only for first or zero order. Otherwise a numerical solution by finite differences, method of lines or finite elements is required. The analytical solution proceeds by the method of separation of variables which converts the PDE into one ODE with variables separable and the other a Bessel equation. The final solution is an infinite series whose development is quite elaborate and should be sought in books on Fourier series or partial differential equations. 

The partial differential equations representing material and energy balances of a reaction in a packed bed are rarely solvable by analytical means, except perhaps when the reaction is of zero or first order. Two examples of derivation of the equations and their analytical solutions are P8.0.1.01 and P8.01.02. In more complex cases analytical, approximations can be made (by "Collocation" or "Perturbation", for instance), but these usually are quite sophisticated to apply. Numerical solutions, on the other hand, are simple in concept and are readily implemented on a computer. Two such methods that are suited to nonlinear kinetics problems will be described. 

The nonlinearity of the system of partial differential equations (51) and (52) poses a serious obstacle to finding an analytical solution. A reported analytical solution for the nonlinear problem of diffusion coupled with complexation kinetics was erroneous [12]. Thus, techniques such as the finite element method [53-55] or appropriate change of variables (applicable in some cases of planar diffusion) [56] should be used to find the numerical solution. One particular case of the nonlinear problem where an analytical solution can be given is the steady-state for fully labile complexes (see Section 3.3). However, there is a reasonable assumption for many relevant cases (e.g. for trace elements such as... 

The process inputs are defined as the heat input, the product flow rate and the fines flow rate. The steady state operating point is Pj =120 kW, Q =.215 1/s and Q =.8 1/s. The process outputs are defined as the thlrd moment m (t), the (mass based) mean crystal size L Q(tK relative volume of crystals vr (t) in the size range (r.-lO m. In determining the responses of the nonlinear model the method of lines is chosen to transform the partial differential equation in a set of (nonlinear) ordinary differential equations. The time responses are then obtained by using a standard numerical integration technique for sets of coupled ordinary differential equations. It was found that discretization of the population balance with 1001 grid points in the size range 0. to 5 10 m results in very accurate solutions of the crystallizer model. 

The resulting set of model partial differential equations (PDEs) were solved numerically according to the method of lines, applying orthogonal collocation techniques to the discretization of the unknown variables along both the z and x coordinates and integrating the resulting ordinary differential equation (ODE) system in time. 

Equations (15)-(23) can be solved numerically using a combination of IMSL and NAG library subroutines. The DIVPAG subroutine of IMSL for ordinary differential equations and the D03PAF subroutine of NAG library for partial differential equations may serve the purpose and we applied this method for simulating the results for ELM extraction of CPC [26] and cephalexin [25]. The agreement between model prediction and experimental data was found to be quite reasonable. 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

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