In the energy balance formulation of the finite difference method, it is recommended that all heat transfer at the boundaries of the volume element be assumed to be into the volume element even for steady heat conduction. Is this a valid recommendation even though it seems to violate the conservation of energy principle?

The finite difference method is a numerical technique to solve partial differential equations by discretizing the problem domain into a grid and solving the equations at the grid points. In the context of heat conduction, the finite difference method is used to find the temperature distribution in a material given the boundary conditions.

In the energy balance formulation of the finite difference method, we consider the energy entering and leaving a volume element, as well as any internal energy generated within the element. The principle of conservation of energy, which states that the total energy in a closed system remains constant, should be maintained throughout the problem domain.

The recommendation states that all heat transfer at the boundaries of the volume element should be assumed as into the volume element, even for steady heat conduction. This means that the energy being transferred across the boundary is always considered as being added to the volume element, irrespective of its direction (inward or outward).

At first glance, this recommendation may seem to violate the conservation of energy principle, as it appears to increase energy within a volume element without accounting for energy leaving the element. However, in the case of steady heat conduction, the rate of heat transfer across the boundaries is constant, and there is no net accumulation or loss of energy within the volume element. Furthermore, considering neighboring volume elements, heat transfer considered as inward for one volume element would also be considered as inward for the neighboring volume element, effectively canceling out and maintaining the conservation of energy in the discretized problem domain. In addition, it's important to note that the finite difference method is a numerical approximation method, and this recommendation is a simplification to ease the computation, which doesn't affect the accuracy of the solution when convergence is met.

In conclusion, while the recommendation to assume all heat transfer at the boundaries of volume element as inward heat transfer may seem to violate the conservation of energy principle, it is indeed valid for the energy balance formulation of the finite difference method even for steady heat conduction. This simplification does not negatively impact the overall energy balance or accuracy of the numerical solution, as any potential imbalance is compensated for in the discretized problem domain, maintaining the conservation of energy.