Finite difference methods explicit method

In the finite difference method an explicit technique would evaluate the right-hand side at the /ith time level. 

W. Herrmann and L.D. Bertholf, Explicit Lagrangian Finite-Difference Methods,... 

A numerical method to simulate the performance of the storage with the PCM module was implemented using an explicit finite-difference method. The discretization of the model can be seen in Figure 145. 

Equation (5.62) for the current-potential response in CV has been deduced by assuming that the diffusion coefficients of species O and R fulfill the condition Do = >r = D. If this assumption cannot be fulfilled, this equation is not valid since in this case the surface concentrations are not constant and it has not been possible to obtain an explicit solution. Under these conditions, the CV curves corresponding to Nemstian processes have to be obtained by using numerical procedures to solve the diffusion differential equations (finite differences, Crank-Nicholson methods, etc. see Appendix I and ([28])3. 

The effectiveness factors and n, defined as the ratios of the actual reaction rates at time 0 to the maximum reaction rates on a clean catalyst, are obtained nEmerically from equations [4] -[9]. An explicit finite difference method was used to solve the partial differential equations without further simplifications. Densities, porosities and clean catalyst pore diameters were measured experimentally. The maximum coke content is assumed to be that which fills the pore completely. The tortuosity is taken as 2.3, as discussed by Satterfield et al. (14). 

In the last section we considered explicit expressions which predict the concentrations in elements at (t + At) from information at time t. An error is introduced due to asymmetry in relation to the simulation time. For this reason implicit methods, which predict what will be the next value and use this in the calculation, were developed. The version most used is the Crank-Nicholson method. Orthogonal collocation, which involves the resolution of a set of simultaneous differential equations, has also been employed. Accuracy is better, but computation time is greater, and the necessity of specifying the conditions can be difficult for a complex electrode mechanism. In this case the finite difference method is preferable7. 

Eqs. (H) through (L) will be solved by the explicit finite difference method. Substitute... 

Considering both radiation and convection heal transfers and using the explicit finite difference method with a mesh size of iiv - 2.5 cm and a time step of dr = 5 min, determine the temperatures of tlie inner and outer surfaces of the roof at 6 am. Also, determine the average rale of heat transfer through the roof during that night. 

A.r = 2 cm and a time step to be Ar = 0.5 s, determine the nodal temperatures after 5 rain by using (he explicit finite difference method. Also, determine how long it will take for steady conditions to be reached. 

The explicit, finite difference method (9,10) was used to generate all the simulated results. In this method, the concurrent processes of diffusion and homogeneous kinetics can be separated and determined independently. A wide variety of mechanisms can be considered because the kinetic flux and the diffusional flux in a discrete solution "layer" can be calculated separately and then summed to obtain the total flux. In the simulator, time and distance increments are chosen for convenience in the calculations. Dimensionless parameters are used to relate simulated data to real world data. 

Example 2.6 A steel plate with the material properties A = 15.0 W/Km and a = 3.75 10-6 m2/s is 28 = 270mm thick and has a constant initial temperature o At time to the plate is brought into contact with a fluid which has a temperature s < o that is constant with respect to time. The heat transfer coefficient at both surfaces of the plate is a = 75W/m2K. The temperatures during the cooling of the plate are to be numerically determined. Simple initial and boundary conditions were intentionally chosen, so that the accuracy of the finite difference method could be checked when compared to the explicit solution of the case dealt with in section 2.3.3. 

This robust higher order finite-difference method, originally presented in [10,13,25], develops a seven-point spatial operator along with an explicit six-stage time-advancing technique of the Runge-Kutta form. For the former operator, two central-difference suboperators are required a) an antisymmetric... 

S. D. Gedney, J. A. Roden, N. K. Madsen. A. H. Mohammadian, W. F. Hall, V. Sankar, and C. Rowell, Explicit time-domain solutions of Maxwell s equations via generalized grids, in Advances in Computational Electrodynamics The Finite-Difference Time-Domain Method, A. Taflove, Ed. Norwood, MA Artech House, 1998, ch. 4, pp. 163—262. 

Finite difference methods have been used bpth to test the assumptions made in the derivation of eqn. (27) under the Leveque approximation [35] and to solve electrochemical diffusion-kinetic problems with the full parabolic profile [36-38]. The suitability of the various finite difference methods commonly encountered has been thoroughly investigated by Anderson and Moldoveanu [37], who concluded that the backward implicit (BI) method is to be preferred to either the simple explicit method [39] or the Crank-Nichol-son implicit method [40]. 

Initially (time zero) the values of c are known and therefore the values of cj (concentrations at t = At) can be calculated by direct application of (25.93). This approach can be repeated and the values of c"+1 can be calculated from the previously calculated values of c . This is an example of an explicit finite difference method, where, if approximate solution values are known at time t = nAt, then approximate values at time tn+1 = (n+ 1) At may be explicitly and immediately calculated using (25.93). Typically, explicit techniques require that constraints be placed on the size of Af that may be used to avoid significant numerical errors and for stable operation. In a stable method, unavoidable small errors in the solution are suppressed with time in an unstable method a small initial error may increase significandy, leading to erroneous results or a complete failure of the method. Equation (25.93) is stable only if At < (Ax)2/2, and therefore one is obliged to use small integration timesteps. 

Now, even if values of c" i = 1,2,..., ) are known, (25.95) cannot be solved explicitly, as c"+l is also a function of the unknown c, 1 and c" /. However, all k equations of the form (25.95) for i = 1,2,..., k form a system of linear algebraic equations with k unknowns, namely, c"+l, c"+l,..., c"+l. This system can be solved and the solution can be advanced from r to tn+. This is an example of an implicit finite difference method. In general, implicit techniques have better stability properties than explicit methods. They are often unconditionally stable and any choice of Ar and Ax may be used (the choice is ultimately based on accuracy considerations alone). 

Calculation was made by using a finite difference method in an explicit form. 

In Fig. 9.4 we plot the corresponding bifurcation diagram. A forward or supercritical bifurcation occurs at T = L. We depict with symbols the values obtained by integrating (9.1) numerically using an explicit finite difference method with a... 

Output from Program 2 for the Explicit Finite-Difference Method... 

There are many numerical approaches one can use to approximate the solution to the initial and boundary value problem presented by a parabolic partial differential equation. However, our discussion will focus on three approaches an explicit finite difference method, an implicit finite difference method, and the so-called numerical method of lines. These approaches, as well as other numerical methods for aU types of partial differential equations, can be found in the literature [5,9,18,22,25,28-33]. 

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