Explicit, Central Difference Solutions

The choice of time discretization is determined by preference and the complexity of the equation to be solved. Explicit differences are simple and do not require iteration, but they can require small time steps for an accurate solution. Imphcit and fully implicit differences allow for accurate solutions at significantly larger time steps but require additional computations through the iterations that are normally required. 

Flux rate - Flux rate + Source - Sink = Accumulation IN OUT rate rate 

We could perform our discretization on the governing partial differential equations, which are different for each case, but there is no need to go beyond the most fundamental equation of mass transport, equation (2.1). 

Central differences are typically used when the flux is predominately due to a diffusive term, where the gradients on both sides of the control volume are important. An exphcit equation is one where the unknown variable can be isolated on one side of the equation. To get an explicit equation in a computational routine, we must use the flux and source/sink quantities of the previous time step to predict the concentration of the next time step. 

We will use the diffusive flux term of equation (2.2), applied at the i+1 / 2 interface between the (i, j, k) control volume and the (i + 1,k) control volume  

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EXAMPLE 7.1 Unsteady dissolution of ammonia into groundwater (unsteady, onedimensional solution with pulse boundary conditions) solved with an explicit, central difference computation... 

Solve the tanker truck spill problem of Example 2.2 using explicit, central differences to predict concentrations over time in the groundwater table. Compare these with those of the analytical solution. The mass spilled is 3,000 kg of ammonia over 100 nf, and the effective dispersion coefficient through the groundwater matrix is 10 m2/s. 

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

The flux terms are discretized at the cell interface. Because diffusion occurs in both directions, we will use central differences. In addition, we will use an explicit solution technique by discretizing our flux terms at only the n time step ... 

In section 3.1.4, an analytical series solution using the matrizant was developed for the case where the coefficient matrix is a function of the independent variable. This methodology provides series solutions for Boundary value problems without resorting to any conventional series solution technique. In section 3.1.5, finite difference solutions were obtained for linear Boundary value problems as a function of parameters in the system. The solution obtained is equivalent to the analytical solution because the parameters are explicitly seen in the solution. One has to be careful when solving convective diffusion equations, since the central difference scheme for the first derivative produces numerical oscillations. 

Coefficients a, a2 and b are obtained by the Fourier analysis and the relatively rapid solution of the resulting tridiagonal system of equations, due to the implicit nature of (2.31). A typical set is a = 22, a% = 1, and b = 24. To comprehend their function, let us observe Figure 2.2 that assumes the computation of dHy/dx and dEz/dx at i = 0. For the first case, constraint Ey = Ez = Hx = 0 at i = 0 indicates that dHy/dx (likewise for all H derivatives) must also be zero. In the second case, to calculate dEz/dx at i = one needs its values at i = —, . Nonetheless, point i = — is outside the domain and to find a reliable value for the tridiagonal matrix, the explicit, sixth-order central-difference scheme is selected... 

For explicit solution we replace the axial derivatives with a forward-difference approximation, and the radial derivatives with central differences. Consider Figure 7.37, where the closed circles represent the grid points with known values of everything that will be used to determine the new point, shown as the open circle. The numerical setup is as follows... 

The finite volume method based on central differences is not dissipative. Thus, high-frequency oscillations of error are not damped near discontinuities, and the procedure cannot converge to the solution in steady state. To eliminate these spurious oscillations, dissipative terms can be introduced explicitly through artificial dissipation [1,19]. 

Courant condition when using an explicit time integration method (e.g., central difference method) for such small elements leads to small time increments and a huge computational cost. To resolve this problem, the Newmark-/ method (jS = 1/4, 5 = 1/2) is used for the time integration. Note that an explicit scheme is selected to compute the response robustly an implicit scheme sometimes does not reach an accurate solution because of complex nonlinear constitutive relation. Thus, to verify the convergence of a solution, attention must be paid to the spatial and temporal discretization. 

Implicit methods Let us now consider some implicit methods for solution of parabolic equations. We utilize the grid of Fig. 6.7, in which the half point in the f-direction (/, n + V2) is shown. Instead of expressing du/dt in terms of forward difference around (/, n), as it was done in the explicit form, we express this partial derivative in terms of central difference around the half point ... 

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