Finite differencing scheme

In the present simulations, the mass of beads is taken as unit mass, and m = 0.5 for mass of all other ions. Friction coefficient is chosen as constant 1.0r , where t is the time unit of the system. The velocity Verlet finite-differencing scheme is chosen for the integration of Eq. (6.7). By fixings, s, and k T the salt concentration is varied from salt-free to high salt. The monomer density of the solution is varied from 1 X 10 to 3 X The chain lengths considered are ofN — 20,40,60, and... 

Not only can the tm-bulence model influence the simulated flow pattern significantly, but also the finite differencing scheme. In Fig. 7.1.3 profiles of axial velocity from simulations with a first order and a second order differencing scheme are compared (Phyfe, 1999). Two different turbulence models. 

Fig. 7.1.3. Comparison of axial velocity profiles by CFD using first and second order finite differencing schemes and two different turbulence models (Phyfe, 1999)...

Central differences were used in Equation (5.8), but forward differences or any other difference scheme would suffice as long as the step size h is selected to match the difference formula and the computer (machine) precision with which the calculations are to be executed. The main disadvantage is the error introduced by the finite differencing. 

P 61] The numerical simulations were based on the solution of the incompressible Navier-Stokes equation and a convection-diffusion equation for a concentration field by means of the finite-volume method [152], The Einstein convention of summation over repeated indices was used. For pressure-velocity coupling, the SIMPLEC algorithm and for discretization of the species concentration equation the QUICK differencing scheme were applied. Hybrid and the central differencing schemes referred to velocities and pressure, respectively (commercial flow solvers CFX4 and CFX5). 

To illustrate the principles of the finite volume method, as a first approach, the implicit upwind differencing scheme is used for a multi-dimensional problem. Although the upwind differencing scheme is very diffusive, this scheme is frequently recommended on the grounds of its stability as the preferred method for treatment of convection terms in multiphase flow and determines the basis for the implementation of many higher order upwinding schemes. 

The mass-weighted Cartesian coordinates are denoted Using Eq. (3), the numerical value of the Blk matrix elements are determined for any value of the internal coordinates via a finite difference scheme. Once the B-matrix is constructed, expansions for Gtj and V, about the equilibrium configuration of the molecule, can be evaluated using higherordering differencing techniques. 

At each time step, values of the variables are calculated at every i, and these are used as input for the next time stepping. This scheme of discretization is known as Lax-Friedrichs finite difference scheme, which is first order accurate [22,23]. In order to ensure stability during time stepping, the variables at time n are approximated as the average of their values at (i - l)th and (f+l)th nodes instead of simple forward differencing, that is, (cm)" -(cm) or According to the Courant-... 

Prominent representatives of the first class are predictor-corrector schemes, the Runge-Kutta method, and the Bulir-sch-Stoer method. Among the more specific integrators we mention, apart from the simple Taylor-series expansion of the exponential in equation (57), the Cayley (or Crank-Nicholson) scheme, finite differencing techniques, especially those of second or fourth order (SOD and FOD, respectively) the split-operator, method and, in particular, the Chebychev and the shoit-time iterative Lanczos (SIL) integrators. Some of the latter integration schemes are norm-conserving (namely Cayley, split-operator, and SIL) and thus accumulate only... 

Finite difference schemes based on the principles above are called central differencing . Central differencing schemes have stability problems in situations where convective transport is significant compared to the diffusive transport. Various methods have been devised to overcome this problem. The upwind scheme assigns the value of at a cell boundary to be the value at the node point from which the fluid is flowing, rather than the mean as in Eqs. (7.1.5) and (7.1.6). 

In addition to the central differencing and upwind differencing schemes, which are first-order schemes, another popular finite difference scheme is the QUICK scheme, a second-order upwind differencing scheme. Higher order means that more node points are involved when estimating the values of the dependent variables and their derivatives for formulating the finite difference equations. 

Numerical analysis. Equation 21-76 is conveniently solved, again using finite difference time-marching schemes. We always central difference our first and second-order space derivatives, while backward differencing in time, with respect to the nodal point (ri,tn). Furthermore, we will evaluate all nonlinear saturation-dependent coefficients at their previous values in time. This leads to... 

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