Finite difference methods discretization

Finite difference method discretizes the function on rectangular grids with equal or varying spacing. The function on a node is related to the values on the neighboring nodes which define the grid system. Spatial or time derivatives are approximated by the difference of the values on the neighboring notes or the successive time steps. The... 

For the determination of the approximated solution of this equation the finite difference method and the finite element method (FEM) can be used. FEM has advantages because of lower requirements to the diseretization. If the material properties within one element are estimated to be constant the last term of the equation becomes zero. Figure 2 shows the principle discretization for the field computation. 

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. 

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... 

Finite element methods [20,21] have replaced finite difference methods in many fields, especially in the area of partial differential equations. With the finite element approach, the continuum is divided into a number of finite elements that are assumed to be joined by a discrete number of points along their boundaries. A function is chosen to represent the variation of the quantity over each element in terms of the value of the quantity at the boundary points. Therefore a set of simultaneous equations can be obtained that will produce a large, banded matrix. 

From a physical point of view, the finite difference method is mostly based based on the further replacement of a continuous medium by its discrete model. Adopting those ideas, it is natural to require that the principal characteristics of a physical process should be in full force. Such characteristics are certainly conservation laws. Difference schemes, which express various conservation laws on grids, are said to be conservative or divergent. For conservative schemes the relevant conservative laws in the entire grid domain (integral conservative laws) do follow as an algebraic corollary to difference equations. 

A numerical method to simulate the performance of the storage with the PCM module was implemented using an explicit finite-difference method. The discretization of the model can be seen in Figure 145. 

The Finite Difference Method (FD)168 169. This is a general method applicable for systems with arbitrary chosen local dielectric properties. In this method, the electrostatic potential (RF) is obtained by solving the discretized Poisson equation ... 

The governing dimensionless partial derivative equations are similar to those derived for cyclic voltammetry in Section 6.2.2 for the various dimerization mechanisms and in Section 6.2.1 for the EC mechanism. They are summarized in Table 6.6. The definition of the dimensionless variables is different, however, the normalizing time now being the time tR at which the potential is reversed. Definitions of the new time and space variables and of the kinetic parameter are thus changed (see Table 6.6). The equation systems are then solved numerically according to a finite difference method after discretization of the time and space variables (see Section 2.2.8). Computation of the... 

The system of hyperbolic and parabolic partial differential equations representing the ID or 2D model of monolith channel is solved by the finite differences method with adaptive time-step control. An effective numerical solution is based on (i) discretization of continuous coordinates z, r and t, (ii) application of difference approximations of the derivatives, (iii) decomposition of the set of equations for Ts, T, c and cs, (iv) quasi-linearization of... 

Most of these tools use the finite difference method (at least for one-dimensional models) in which the continuous space coordinate is divided into a number of boxes. So we are back to the box-model technique. To demonstrate the procedure, in Box 23.4 we show how the partial differential Eqs. 23-44 and 23-45 are transformed into discrete (box) equations. 

The analysis of polymer processing is reduced to the balance equations, mass or continuity, energy, momentum and species and to some constitutive equations such as viscosity models, thermal conductivity models, etc. Our main interest is to solve this coupled nonlinear system of equations as accurately as possible with the least amount of computational effort. In order to do this, we simplify the geometry, we apply boundary and initial conditions, we make some physical simplifications and finally we chose an appropriate constitutive equations for the problem. At the end, we will arrive at a mathematical formulation for the problem represented by a certain function, say / (x, T, p, u,...), valid for a domain V. Due to the fact that it is impossible to obtain an exact solution over the entire domain, we must introduce discretization, for example, a grid. The grid is just a domain partition, such as points for finite difference methods, or elements for finite elements. Independent of whether the domain is divided into elements or points, the solution of the problem is always reduced to a discreet solution of the problem variables at the points or nodal pointsinxxnodes. The choice of grid, i.e., type of element, number of points or nodes, directly affects the solution of the problem. 

The finite difference method (FDM) is probably the easiest and oldest method to solve partial differential equations. For many simple applications it requires minimum theory, it is simple and it is fast. When a higher accuracy is desired, however, it requires more sophisticated methods, some of which will be presented in this chapter. The first step to be taken for a finite difference procedure is to replace the continuous domain by a finite difference mesh or grid. For example, if we want to solve partial differential equations (PDE) for two functions 4> x) and w(x, y) in a ID and 2D domain, respectively, we must generate a grid on the domain and replace the functions by functions evaluated at the discrete locations, iAx and jAy, (iAx) and u(iAx,jAy), or 4>i and u%3. Figure 8.1 illustrates typical ID and 2D FDM grids. 

The independent variable in ordinary differential equations is time t. The partial differential equations includes the local coordinate z (height coordinate of fluidized bed) and the diameter dp of the particle population. An idea for the solution of partial differential equations is the discretization of the continuous domain. This means discretization of the height coordinate z and the diameter coordinate dp. In addition, the frequently used finite difference methods are applied, where the derivatives are replaced by central difference quotient based on the Taylor series. The idea of the Taylor series is the value of a function f(z) at z + Az can be expressed in terms of the value at z. 

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. 

Newton variants are constructed by combining various strategies for the individual components above. These involve procedures for formulating Hk or Hk, dealing with structures of indefinite Hessians, and solving for the modified Newton search direction. For example, when Hk is approximated by finite differences, the discrete Newton subclass emerges.5 91-94 When Hk, or its inverse, is approximated by some modification of the previously constructed matrix (see later), QN methods are formed.95-110 When is nonzero, TN methods result,111-123 because the solution of the Newton system is truncated before completion. 

For the solution of sophisticated mathematical models of adsorption cycles including complex multicomponent equilibrium and rate expressions, two numerical methods are popular. These are finite difference methods and orthogonal collocation. The former vary in the manner in which distance variables are discretized, ranging from simple backward difference stage models (akin to the plate theory of chromatography) to more involved schemes exhibiting little numerical dispersion. Collocation methods are often thought to be faster computationally, but oscillations in the polynomial trial function can be a problem. The choice of best method is often the preference of the user. 

In order to solve the first principles model, finite difference method or finite element method can be used but the number of states increases exponentially when these methods are used to solve the problem. Lee et u/.[8] used the model reduction technique to reslove the size problem. However, the information on the concentration distribution is scarce and the physical meaning of the reduced state is hard to be interpreted. Therefore, we intend to construct the input/output data mapping. Because the conventional linear identification method cannot be applied to a hybrid SMB process, we construct the artificial continuous input/output mapping by keeping the discrete inputs such as the switching time constant. The averaged concentrations of rich component in raffinate and extract are selected as the output variables while the flow rate ratios in sections 2 and 3 are selected as the input variables. Since these output variables are directly correlated with the product purities, the control of product purities is also accomplished. 

S-99C Explain why the local discretization error of the finite difference method is proportional to the square of the step size. 

To illustrate the validity of the models presented in the previous section, results of validation experiments using lab-scale BSR modules are taken from Ref. 7. For those experiments, the selective catalytic reduction (SCR) of nitric oxide with excess ammonia served as the test reaction, using a BSR filled with strings of a commercial deNO catalyst shaped as hollow extrudates (particle diameter 1.6 or 3.2 mm). The lab-scale BSR modules had square cross sections of 35 or 70 mm. The kinetics of the model reaction had been studied separately in a recycle reactor. All parameters in the BSR models were based on theory or independent experiments on pressure drop, mass transfer, or kinetics none of the models was later fitted to the validation experiments. The PDFs of the various models were solved using a finite-difference method, with centered differencing discretization in the lateral direction and backward differencing in the axial direction the ODEs were solved mostly with a Runge-Kutta method [16]. The numerical error of the solutions was... 

The explicit, finite difference method (9,10) was used to generate all the simulated results. In this method, the concurrent processes of diffusion and homogeneous kinetics can be separated and determined independently. A wide variety of mechanisms can be considered because the kinetic flux and the diffusional flux in a discrete solution "layer" can be calculated separately and then summed to obtain the total flux. In the simulator, time and distance increments are chosen for convenience in the calculations. Dimensionless parameters are used to relate simulated data to real world data. 

In the finite difference method, the derivatives dfl/dt, dfl/dx and d2d/dx2, which appear in the heat conduction equation and the boundary conditions, are replaced by difference quotients. This discretisation transforms the differential equation into a finite difference equation whose solution approximates the solution of the differential equation at discrete points which form a grid in space and time. A reduction in the mesh size increases the number of grid points and therefore the accuracy of the approximation, although this does of course increase the computation demands. Applying a finite difference method one has therefore to make a compromise between accuracy and computation time. 

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. 

Finite difference methods (FDM) are directly derived from the space time grid. Focusing on the space domain (horizontal lines in Fig. 6.6), the spatial differentials are replaced by discrete difference quotients based on interpolation polynomials. Using the dimensionless formulation of the balance equations (Eq. 6.107), the convection term at a grid point j (Fig. 6.6) can be approximated by assuming, for example, the linear polynomial. 

The selected mathematical model is represented by a discretization method for approximating the differential equations by a system of algebraic equations for the variables at some set of discrete locations in space and time. Many different approaches are used in reactor engineering , but the most important of them are the simple finite difference methods (FDMs), the flrrx conservative finite volume methods (FVMs), and the accurate high order weighted residual methods (MWRs). 

The finite approximations to be used in the discretization process have to be selected. In a finite difference method, approximations for the derivatives at the grid points have to be selected. In a finite volume method, one has to select the methods of approximating surface and volume integrals. In a weighted residual method, one has to select appropriate trail - and weighting functions. A compromise between simplicity, ease of implementation, accuracy and computational efficiency has to be made. For the low order finite difference- and finite volume methods, at least second order discretization schemes (both in time and space) are recommended. For the WRMs, high order approximations are normally employed. 

A consistent numerical scheme produces a system of algebraic equations which can be shown to be equivalent to the original model equations as the grid spacing tends to zero. The truncation error represents the difference between the discretized equation and the exact one. For low order finite difference methods the error is usually estimated by replacing all the nodal values in the discrete approximation by a Taylor series expansion about a single point. As a result one recovers the original differential equation plus a remainder, which represents the truncation error. 

The finite volume method has become a very popular method of deriving discretizations of partial differential equations because these schemes preserve the conservation properties of the differential equation better than the schemes based on the finite difference method. 

The first task is to discretize the domain. The computational domain is subdivided into a finite number of small grid cell volumes (GCVs) by a grid which defines both the grid cell volume boundaries and the computational nodes. Note that in the finite difference method, the grid defines the location of the computational nodes solely. 

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