Central Finite Differences

As their name implies, central finite differences are centered at the pivot position and are evaluated utilizing the values of the function to the right and to the left of the pivot position, but located only h/2 distance from it. 

Consider the series of values used in the previous two sections, but with the additional values at the midpoints of the intervals 

Higher-order central differences are similarly derived  

Consistent with the other finite differences, the central finite differences also have coefficients that correspond to those of the binomial expansion (a - bf. Therefore, the general formula of the nth-order central finite difference can be expressed as 

The averager operator shifts its operand by a half interval to the right of the pivot and by a half interval to the left of the pivot, evaluates it at these two positions, and averages the two values. 

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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... 

II. Stable high-order central finite difference schemes on composite adaptive grids with sharp shock resolution,... 

In the Verlet method, this equation is written by using central finite differences (see Interpolation and Finite Differences ). Note that the accelerations do not depend upon the velocities. 

In this Eq. (Js)n is the Jacobi matrix for the solid phase, which contains the derivatives of the mass residuals for the particulate phase to the solid volume fraction. Explicit expressions for the elements of the Jacobi matrix can be obtained from the continuity for the solid phase and the momentum equations. For example for the central element, the following expression is obtained from the solid phase continuity equation, in which the convective terms are evaluated with central finite difference expressions ... 

Gradient diffusion was assumed in the species-mass-conservation model of Shir and Shieh. Integration was carried out in the space between the ground and the mixing height with zero fluxes assumed at each boundary. A first-order decay of sulfur dioxide was the only chemical reaction, and it was suggested that this reaction is important only under low wind speed. Finite-difference numerical solutions for sulfur dioxide in the St. Louis, Missouri, area were obtained with a second-order central finite-difference scheme for horizontal terms and the Crank-Nicolson technique for the vertical-diffusion terms. The three-dimensional grid had 16,800 points on a 30 x 40 x 14 mesh. 

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. 

Using 4 grid points to represent d/dx Instead of using the first order (backward or forward) or the second order (central) finite difference approximation for the first derivative, let us calculate the derivative using four grid points (see Fig. 8.2)... 

Note that Pg > 2 is critical, because the solution presents a sign change, which means the solution becomes unstable (see Figure 8.17). The root of the problem is explained by the info-travel concept. To generate the difference equation (eqn. (8.66)) we used a central finite difference for the convective derivative, which is incorrect, because the information of the convective term cannot travel in the upstream direction, but rather travels with the velocity ux. This means that to generate the FD equation of a convective term, we only take points that are up-stream from the node under consideration. This concept is usually referred to as up-winding technique. For low Pe the solution is stable because diffusion controls and the information comes from all directions. 

Fig 16. NSC for protein and mRNA copies for different size perturbations in kr when time-averaged concentrations are used. First-order (forward or backward) and central finite difference approximations are used. The deterministic NSC for both species is 1 (vertical dashed line). 

Fig. 44 (a) Upwind, (b) downwind and (c) central finite differences. The grey nodes indicate those used in fomnulating the finite difference. 

If this is approximated using (central) finite differences it becomes (105),... 

In this woric, discretisation of both space and time derivatives was executed, based on either central finite difference (CFD) or orthogonal collocation cm finite elements (OCFE) discretisation in the spatial domain and backward finite difference (BFD) discretisation in the time domain. 

X2 = 0.0. The PDEs were discretized using central Finite Differences, with 250 nodes, resulting in a system of 500 algebraic equations. We chose a single decision variable (namely Da) optimisation problem here in order to better characterise our... 

If we subtract Eq. 10.65 from Eq. 10.64, and divide the result by Ax, we obtain the expression of the central finite difference for the first-order differential ... 

Thus, when we replace a partial differential term by a central finite difference, we make an error that is of the order of 0 Ax ), and the coefficient of this error contribution is the third-order partial differential, fi G/dx. ... 

In conclusion, when we replace the first- and second-order partial differential terms in a partial differential equation by central finite difference terms, we make errors that are of the order of 0 Ax ). For most practical purposes, this second-order error is negfigible. By contrast, when we replace the first-order partial differential terms with a forward or a backward finite difference term, we make errors that are of the order of 0 Ax). This first-order error contribution is never negligible. 

As an example, if we select a central finite difference for the first term in the LHS of Eq. 10.61, a backward finite difference term for its second LHS term, and a... 

To attain the fourth-order spatial discretization of (2.15) and (2.16), the following central finite-difference schemes are employed. The former is the Yee s staggered-grid arrangement, while the latter is the collocated-mesh configuration, where component f and its derivatives are positioned at the same node. So,... 

For the discretization of the 2M jump conditions, the algorithm uses central finite-difference schemes at each mesh node to construct 2M x 2M algebraic equations. For instance, let us consider jump condition... 

Temporal differentiation, on the other hand, is approximated by central finite-difference (< /, 2ti <52) and central average (/xt, /i2t) operators defined in Table 5.1 [21]. 

The model equations were solved numerically by discretizing the partial differential equations (PDEs) with respect to the spatial coordinate (x). Central finite difference formulae were used to approximate the first and second derivatives (e.g. dc,/dx, d77ck). Thus the PDEs were transformed to ODEs with respect to the reaction time and the finite difference method was used in the numerical solution. The recently developed software of Buzzi Ferraris and Manca was used, since it turned out to be more rapid than the classical code of Hindmarsh. 

If a central finite-difference approximation is used to represent the spatial derivatives in equation (5-20), the following is obtained for point n... 

Related finite-difference techniques have been studied in a number of contexts involving fluids with hard-core [111] and soft potentials [112-116]. The central finite-difference approximation, analyzed systematically in [115], was implemented... 

To solve the model A, the solid phase mass balance equation was discretized using second order central finite differences. For both models, the differential-algebraic equations were solved using a Backward-Difference Formula (BDF algorithm). 

Multistep methods can be used for integration of the system of equations in time, such as Runge-Kutta, and the central finite difference scheme for the spatial approximation of first-order derivatives for each grid point (i,j, k) as shown in Figure 6.4 for uniform meshes by simplicity... 

Model Eq. (6.76) can be discretized in space using a central finite difference scheme of the second order, for a node-centered mesh, as shown in Figure 6.5, as... 

Response sensitivity computation can be performed using different methods, such as the forward/backward/central Finite Difference Method (FDM) (Kleiber et al. 1997, Conte et al. 2003, 2004), the Adjoint Method (AM) (Kleiber et al. 1997), the Perturbation Method (PM) (Kleiber Hien 1992), and the Direct Differentiation Method... 

The discretization scheme, which leads to an error 0 h ) for second-order differential equations (without first derivative) with the lowest number of points in the difference equation, is the method frequently attributed to Nu-merov [494,499]. It can be efficiently employed for the transformed Poisson Eq. (9.232). In this approach, the second derivative at grid point Sjt is approximated by the second central finite difference at this point, corrected to order h, and requires values at three contiguous points (see appendix G for details). Finally, we obtain tri-diagonal band matrix representations for both the second derivative and the coefficient function of the differential equation. The resulting matrix A and the inhomogeneity vector g are then... 

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