Finite Differences method 2

The finite difference method is a commonly used iterative technique that allows numerical solution of complex equations. In particular, forward difference methods approximate the solutions to differential equations by replacing derivative expressions with approximately equivalent difference quotients (Dusinberre, 1961).
In the case of heat transfer through the layers of a road pavement, it is assumed that each layer has uniform properties and is of infinite horizontal extent in order to reduce the problem to the one-dimensional case of determining temperatures at any particular depth and time (see Figure 1).

Figure 1: Finite differenced method
As shown in Figure 111, the domain is portioned in space using a uniform vertical mesh of size Δz and in time using a uniform mesh of size Δt. For each layer, the values of material parameters such as density ρ [kg m^-3], thermal conductivity k [J m^-1 s^-1 K^-1] and specific heat c [J kg^-1 K^-1] are known. These values define the values of conductance K_ij (relative to the material between two points of the mesh) and heat capacity C_i (relative to the material around each mesh point) that are then used in the following equations to calculate the new temperatures T’_i from the current temperatures T_i:
                                                                                          (1)
where
with j ≠ i                                                                                                                (2)
with j ≠ i                                                                                                                   (3)
                                                                                                                               (4)
Q_gi is the energy that enters the system at the node i. In our case, only the surface node receives the energy Q_g (that has been calculated in the previous sections), therefore this term is zero for all the internal nodes of the mesh.
A consideration for equations 1-4 defines a criterion for the stability of this method. It can be seen, in fact, that if Δt is chosen large enough, the coefficients F_ii could become negative. This would be absurd because it would mean that the higher the temperature T_i is at the current time step the lower it is going to be at the next. In order to avoid this situation, therefore, the size of the time step has to comply with the following rule in every region of the system:
                                                                                                                                         (5)
which effectively states that the maximum acceptable value for Δt is directly proportional to the square of Δz.

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