Solution scheme development of the nonhomogeneous heat conduction equation in cylindrical coordinates with Neumann boundary conditions by finite difference method

Partial differential heat conduction equations are typically used to determine temperature distribution within any solid domain. The difficulty and complexity of the solution of the equation depend on differential equation characteristics, boundary conditions, coordinate systems, and the number of dependent variables. In the current study, the numerical solution schemes were developed by the Explicit Finite Difference and the Implicit Method- the Crank-Nicolson techniques for the partial differential heat conduction equation including heat generation term described as one-dimensional, time-dependent with the Neumann boundary conditions. The solution schemes were, then, applied to the battery problem including highly varying heat generation. Besides, the solution of the problem was performed by using Matlab pdepe solver to verify the developed schemes. Results suggest that the Crank-Nicolson scheme is unconditionally stable, whereas the explicit scheme is only stable when the Courant-Friedrichs-Lewy condition requirement is less than 0.3404. Comparing the developed schemes to the results obtained from the pdepe solver, the schemes are as reliable as the pdepe solver with certain grid structures. Besides, the developed numerical schemes allow for shorter computational times than the pdepe solver at the same grid structures when considering CPU times.