Spatial finite differences

First order hyperbolic differential equations transmit discontinuities without dispersion or dissipation. Unfortunately, as Carver (10) and Carver and Hinds (11) point out, the use of spatial finite difference formulas introduces unwanted dispersion and spurious oscillation problems into the numerical solution of the differential equations. They suggest the use of upwind difference formulas as a way to diminish the oscillation problem. This follows directly from the concept of domain of influence. For hyperbolic systems, the domain of influence of a given variable is downstream from the point of reference, and therefore, a natural consequence is to use upstream difference formulas to estimate downstream conditions. When necessary, the unwanted dispersion problem can be reduced by using low order upwind difference formulas. 

The Lagrange interpolation polynomial was used to develop the spatial finite difference formulas used for the distance method of lines calculation. For example, the two point polynomial for the solids flux variable F(t,z) can be expressed by... 

Different combinations of spatial finite difference formulas were tried to determine the best set for our system of equations. The two point upwind formula was found to be best for the solids component molar fluxes. The low order formula was used because most of the gasifier reactions turn off abruptly when a component disappears and this creates sharp discontinuities. Higher order formulas tend to flatten out discontinuities, and in some cases, this causes material balances to be lost which then leads to numerical instability problems. Maintaining component material balance is an important aid to preserving numerical stability in the calculations. The low order formulas minimized these difficulties. 

If we divide the airshed into L cells and consider N species, LN ordinary differential equations of the form (15) constitute the airshed model. As might be expected, this model bears a direct relation to the partial differential equations of conservation (7). If we allow the cell size to become small, it can be shown that (15) is the same as the first-order spatial finite difference representation of (7) in which turbulent diffusive transport is neglected—i.e,. 

Solving the heat exchanger equations using spatial finite differences... 

Packages exist that use various discretizations in the spatial direction and an integration routine in the time variable. PDECOL uses B-sphnes for the spatial direction and various GEAR methods in time (Ref. 247). PDEPACK and DSS (Ref. 247) use finite differences in the spatial direction and GEARB in time (Ref. 66). REACOL (Ref. 106) uses orthogonal collocation in the radial direction and LSODE in the axial direction, while REACFD uses finite difference in the radial direction both codes are restricted to modeling chemical reactors. 

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... 

In reality, heat is conducted in all three spatial dimensions. While specific building simulation codes can model the transient and steady-state two-dimensional temperature distribution in building structures using finite-difference or finite-elements methods, conduction is normally modeled one-... 

In the finite-difference appntach, the partial differential equation for the conduction of heat in solids is replaced by a set of algebraic equations of temperature differences between discrete points in the slab. Actually, the wall is divided into a number of individual layers, and for each, the energy conserva-tk>n equation is applied. This leads to a set of linear equations, which are explicitly or implicitly solved. This approach allows the calculation of the time evolution of temperatures in the wall, surface temperatures, and heat fluxes. The temporal and spatial resolution can be selected individually, although the computation time increa.ses linearly for high resolutions. The method easily can be expanded to the two- and three-dimensional cases by dividing the wall into individual elements rather than layers. 

The partial differential equations describing the catalyst particle are discretized with central finite difference formulae with respect to the spatial coordinate [50]. Typically, around 10-20 discretization points are enough for the particle. The ordinary differential equations (ODEs) created are solved with respect to time together with the ODEs of the bulk phase. Since the system is stiff, the computer code of Hindmarsh [51] is used as the ODE solver. In general, the simulations progressed without numerical problems. The final values of the rate constants, along with their temperature dependencies, can be obtained with nonlinear regression analysis. The differential equations were solved in situ with the backward... 

If for example we discretize the region over which the PDE is to be solved into M grid blocks, use of finite differences (or any other discretization scheme) to approximate the spatial derivatives in Equation 10.1 yields the following system of ODEs ... 

Export processes are often more complicated than the expression given in Equation 7, for many chemicals can escape across the air/water interface (volatilize) or, in rapidly depositing environments, be buried for indeterminate periods in deep sediment beds. Still, the majority of environmental models are simply variations on the mass-balance theme expressed by Equation 7. Some codes solve Equation 7 directly for relatively large control volumes, that is, they operate on "compartment" or "box" models of the environment. Models of aquatic systems can also be phrased in terms of continuous space, as opposed to the "compartment" approach of discrete spatial zones. In this case, the partial differential equations (which arise, for example, by taking the limit of Equation 7 as the control volume goes to zero) can be solved by finite difference or finite element numerical integration techniques. 

Models of the above have been presented by various researchers of the U.S. Geological Survey (USGS) and the academia. The above equation has been solved principally (a) numerically over a temporal and spatial discretized domain, via finite difference or finite element mathematical techniques (e.g., 11) (b) analytically, by seeking exact solutions for simplified environmental conditions (e.g., 12) or (c) probabilistically (e.g., 13). 

Model selection, application and validation are issues of major concern in mathematical soil and groundwater quality modeling. For the model selection, issues of importance are the features (physics, chemistry) of the model its temporal (steady state, dynamic) and spatial (e.g., compartmental approach resolution) the model input data requirements the mathematical techniques employed (finite difference, analytic) monitoring data availability and cost (professional time, computer time). For the model application, issues of importance are the availability of realistic input data (e.g., field hydraulic conductivity, adsorption coefficient) and the existence of monitoring data to verify model predictions. Some of these issues are briefly discussed below. 

Commercially available CFD codes use one of the three basic spatial discretization methods finite differences (FD), finite volumes (FV), or finite elements (FE). Earlier CFD codes used FD or FV methods and have been used in stress and flow problems. The major disadvantage of the FD method is that it is limited to structured grids, which are hard to apply to complex geometries and... 

The independent variables on which fJK depends are k and t. The principal advantage of using this formulation is that spatial derivatives become summations over wavenumber space. The resulting numerical solutions have higher accuracy compared with finite-difference methods using the same number of grid points. 

The discretization points and the variables have been defined at, respectively, the centre of the segment and the boundary between two neighbour segments. Standard second-order finite difference approximations were used to discretize the spatial derivatives ... 

Thermal/structural response models are related to field models in that they numerically solve the conservation of energy equation, though only in solid elements. Finite difference and finite element schemes are most often employed. A solid region is divided into elements in much the same way that the field models divide a compartment into regions. Several types of surface boundary conditions are available adiabatic, convection/radiation, constant flux, or constant temperature. Many ofthese models allow for temperature and spatially dependent material properties. 

As stated above, the spatial derivative is approximated without regard to the time level. The distinction between explicit and implicit solutions depends on the time level at which the spatial derivatives are evaluated. Finite-difference stencils for explicit and implicit Euler methods are illustrated in Fig. 4.13. 

Fig. 4.13 Finite-difference stencils for the explicit and implicit Euler methods. The spatial index is j and the time index is n. For equally spaced radial mesh intervals of dr, rj = (j — 1 )dr, 1 < j < J. For equally spaced time intervals, tn = (n — 1 )dt, n > 1.

To analyze this phenomenon further, 2D numerical simulations of (49) and (50) were performed using a central finite difference approximation of the spatial derivatives and a fourth order Runge-Kutta integration of the resulting ordinary differential equations in time. Details of the simulation technique can be found in [114, 119]. The material parameters of the polymer blend PDMS/PEMS were used and the spatial scale = (K/ b )ll2 and time scale r = 2/D were established from the experimental measurements of the structure factor evolution under a homogeneous temperature quench. 

As we can imagine, most of these issues are directly related to the order of the approximation used in the finite difference representation. In fact, the truncation error (as shown in Chapter 7) is the difference between the PDE and the FD representation, which is represented by the terms collapsed in 0(Axn). For problems represented by PDEs with more than one independent variable the truncation error will be the sum of the truncation error for each FD representation. For example, for a transient one dimensional PDE, where we use a first order approximation for the time derivative and a second order for the spatial derivative, we will have a truncation error that is O(At) + 0 Ax2), which can also be written as 0(At, Ax2). 

The most common methodology when solving transient problems using the finite element method, is to perform the usual Garlerkin weighted residual formulation on the spatial derivatives, body forces and time derivative terms, and then using a finite difference scheme to approximate the time derivative. The development, techniques and limitations that we introduced in Chapter 8 will apply here. The time discretization, explicit and implicit methods, stability, numerical diffusion etc., have all been discussed in detail in that chapter. For a general partial differential equation, we can write... 

If we use an implicit finite difference time stepping procedure, we must evaluate the spatial derivatives in the next time step j +1. In order to arrive at a linear set of algebraic equations that can be solved using standard matrix routines, we would the like the final set of equation to be of the form... 

A common method for solving partial differential equations (PDEs) is known as the method of lines. Here, finite difference approximations for spatial derivatives are used to convert a PDE model to a large set of ordinary differential equations, which are then solved using any of the ODE integration techniques discussed earlier. 

The simplest finite-difference representation of the mixed boundary condition (8-44) may be readily obtained by considering for the spatial derivative the forward-... 

Having established the appropriate finite-difference expressions of the spatial derivatives, the diffusion equation may be approximated as follows ... 

In this explicit scheme, the first-order forward difference approximation is used for the time derivative. The second-order central difference approximation is used for the spatial derivatives. Hence, the finite difference equation (FDE) of the partial differential equation (PDE) Eq. (10.2) is... 

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