Discretization of the Equations

Several methods have been employed over the years to solve the Navier-Stokes equations numerically, including the finite difference, finite element, spectral element, and finite volume methods. The focus of this chapter is on the finite volume method, which is described in detail below. Once the method and terminology have been presented, the other methods are discussed briefly in Section 5-3.2.3. 

The parameter F is used to represent the diffusion coefficient for the scalar 4). If 4) is one of the components of velocity, for example, F would represent the viscosity. All sources are collected in the term S. Again, if 4 is one of the components of velocity. S would be the sum of the pressure gradient, the gravitational force, and any other additional forces that are present. The control volume has a node, P, at its center where all problem variables are stored. The transport equation describes the flow of the scalar 4 into and out of the cell through the cell faces. To keep track of the inflow and outflow, the four faces are labeled with lowercase letters representing the east, west, north, and south borders. The neighboring cells also have nodes at their centers, and these are 

This equation can be solved exactly. On a linear domain that extends from x = 0 to X = L, corresponding to the locations of two adjacent cell nodes, with c ) = (ho at X = 0 and cj) = c l at x = L, the solution for cj) at any intermediate location (such as the face) has the form 

The Peclet number, Pe, appearing in this equation is the ratio of the influence of convection to that of diffusion on the flow field  

Depending on the value of the Peclet number, different limiting behavior exists for the variation of h between x = 0 and x = L. These limiting cases are discussed below, along with some more rigorous discretization or differencing schemes that are in popular use today. 

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Before carrying out the discretization of the equations, we have to make a careful mathematical analysis of the problem in order to establish what its most convenient rewriting in order to facilitate the numerical solution. First we observe that between the matrix A, the proper values kj and the (left) proper vectors Zj, we have the equality ZjA = kjZj. Consequently, as a result, the multiplication of Eq. (4.141) by Zj gives ... 

The differential equations presented in Section 5-2 describe the continuous movement of a fluid in space and time. To be able to solve those equations numerically, all aspects of the process need to be discretized, or changed from a continuous to a discontinuous formulation. For example, the region where the fluid flows needs to be described by a series of connected control volumes, or computational cells. The equations themselves need to be written in an algebraic form. Advancement in time and space needs to be described by small, finite steps rather than the infinitesimal steps that are so familiar to students of calculus. All of these processes are collectively referred to as discretization. In this section, disaetiza-tion of the domain, or grid generation, and discretization of the equations are described. A section on solution methods and one on parallel processing are also included. 

S-3.3.5 Numerical Diffusion. Numerical diffusion is a source of error that is always present in finite volume CFD, owing to the fact that approximations are made during the process of discretization of the equations. It is so named because it presents itself as equivalent to an increase in the diffusion coefficient. Thus, in the solution of the momentum equation, the fluid will appear more viscous in the solution of the energy equation, the solution will appear to have a higher conductivity in the solution of the species equation, it will appear that the species diffusion coefficient is larger than in actual fact. These errors are most noticeable when diffusion is small in the actual problem definition. 

One particular problem that massively increases the cost of correlated calculations is the slow convergence of the correlation energy with the size of the basis set that is used for the discretization of the equations. This problem comes about because all commonly used correlation methods try to expand the wavefunction in a linear combination of antisymmetrized orbital products, i.e. Slater determinants. This ansatz, however, cannot correctly describe short-range correlation effects, i.e. the shape of the wavefunction when two electron approach each other closely, and very large expansions augmented with extrapolation techniques are needed to get a sufficiently converged correlation energy. 

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. 

The initial conditions of system (20) coincide with those for the original equations X/,(0) = X" and V/i(0) = V . Appropriate treatments, as discussed in [72], are essential for the random force at large timesteps to maintain thermal equilibrium since the discretization S(t — t ) => 6nml t is poor for large At. This problem is alleviated by the numerical approach below because the relevant discretization of the Dirac function is the inner timestep At rather than a large At. 

The form of the Hamiltonian impedes efficient symplectic discretization. While symplectic discretization of the general constrained Hamiltonian system is possible using, e.g., the methods of Jay [19], these methods will require the solution of a nontrivial nonlinear system of equations at each step which can be quite costly. An alternative approach is described in [10] ( impetus-striction ) which essentially converts the Lagrange multiplier for the constraint to a differential equation before solving the entire system with implicit midpoint this method also appears to be quite costly on a per-step basis. 

Following the discretization of the solution domain Q (i.e. line AB) into two-node Lagrange elements, and representation of T as T = Ni(x)Ti) in terms of shape functions A, (.v), i = 1,2 within the space of a finite element Q, the elemental Galerkin-weighted residual statement of the differential equation is written as... 

In the continuous penalty technique prior to the discretization of the governing equations, the pressure in the equation of motion is substituted from Fquation (3.6) to obtain... 

The use of selectively reduced integration to obtain accurate non-trivial solutions for incompressible flow problems by the continuous penalty method is not robust and failure may occur. An alternative method called the discrete penalty technique was therefore developed. In this technique separate discretizations for the equation of motion and the penalty relationship (3.6) are first obtained and then the pressure in the equation of motion is substituted using these discretized forms. Finite elements used in conjunction with the discrete penalty scheme must provide appropriate interpolation orders for velocity and pressure to satisfy the BB condition. This is in contrast to the continuous penalty method in which the satisfaction of the stability condition is achieved indirectly through... 

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. 

As described in the discrete penalty technique subsection in Chapter 3 in the discrete penalty method components of the equation of motion and the penalty relationship (i.e. the modified equation of continuity) are discretized separately and then used to eliminate the pressure term from the equation of motion. In order to illustrate this procedure we consider the following penalty relationship... 

After substitution of the first- and second-order time derivatives of the unknowns in Equations (4.132) to (4.134) from Equations (4.139) to (4.141) and spatial discretization of the resulting equations in the usual manner the working equations of the scheme are derived. In these equations, fimctions given at time level n+aAt can be interpolated as... 

Standard Galerkin procedure - to discretize the circumferential component of the equation of motion, Equation (5.23), for the calculation of vs. 

Continuous penalty method - to discretize the continuity and (r, z) components of the equation of motion, Equations (5.22) and (5.24), for the calculation of r,. and v. Pressure is computed via the variational recovery procedure (Chapter 3, Section 4). 

Subdivision or discretization of the flow domain into cells or elements. There are methods, called boundary element methods, in which the surface of the flow domain, rather than the volume, is discretized, but the vast majority of CFD work uses volume discretization. Discretization produces a set of grid lines or cuives which define a mesh and a set of nodes at which the flow variables are to be calculated. The equations of motion are solved approximately on a domain defined by the grid. Curvilinear or body-fitted coordinate system grids may be used to ensure that the discretized domain accurately represents the true problem domain. 

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... 

This is possible within the framework of the self-consistent field (SCF) approach to polymer configurations, described more completely elsewhere [18, 19, 51, 52]. Implementation of this method in its full form invariably requires numerical computations which are done in one of two equivalent ways (1) as solutions to diffusion- or Schrodinger-type equations for the polymer configuration subject to the SCF (in which solutions to the continuous-space formulation of the equations are obtained by discretization) or (2) as solutions to matrix equations resulting from a discrete-space formulation of the problem on a lattice. 

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