The Info-Travel Concept

Up to this point, we have discussed how to develop finite difference expressions for PDE s with any desired level of accuracy. We expect that the truncation error, the consistency and stability will be improved as the order of the approximation is increased. However, we have not mentioned the connection between the physics of the problem under consideration, and represented by the PDE, and the way we select the points for the finite difference expression, i.e., i — 2, i — 1, i + 1, or i + 2, as well as the order of the approximation, i.e., first order, second order, etc. In other words, the points, order and grid size are not arbitrarily chosen to approximate the derivatives at a point i, but should be chosen according to the physical principles of the problem. This concept is referred to as info-travel and will be discussed here. 

The balance equations used to model polymer processes have, for the most part, first order derivatives in time, related with transient problems, and first and second order derivatives in space, related with convection and diffusive problems, respectively. Let us take the heat equation over an infinite domain as 

For this diffusive (conductive) problem, any perturbation in the temperature will move, or diffuse, symmetrically along the (x) axis. In other words the information will travel in all directions in space. Therefore, to approximate the second derivatives in space at a point i, we must use points that are symmetrically distributed in space, as shown in Fig. 8.3. 

A good option will be to use a second order approximation for the space derivatives as follows 

To increase the order of the approximation correctly, we must include the same number of points in any direction. This way we are accommodating, as best as possible, the way information travels in this problem. If the choice of points had been asymmetric, even by keeping the second order of the approximation (such as in eqn. (8.27)), the system is actually considering diffusion with a preferred direction, which is not realistic. 

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Following the info-travel concept for the time derivative, we can easily see that it would be unreasonable to use central or forward differences. Such a choice would mean that the derivative of temperature with respect to time at the fcth time step depends on the value of temperature in the future, or the (k + l)th step. Hence, the correct way to express a time derivative should be by using the information in the past only, i.e.,... 

The info-travel concept dictates the selection of the points and the order of the approximation for each individual case. The continuity equation comes from a mass balance on a differential element within the domain. Here, the mass can come and go through all sides of the differential element hence, the information can travel in all directions, requiring a central difference formulation. 

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. 

Here, we are using a second order approximation for the second derivative using the correct info-travel concept for the conduction term. This equation comes from the energy balance within the domain, thus it will be used for the internal nodes n = 2,3 and 4. The boundary condition for the first node is the temperature at the wall to which the fin is attached to... 

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