A comparison of Finite difference schemes to Finite volume scheme for axially symmetric 2D heat equation

Abstract

In this work, we compare finite difference schemes to finite volume scheme for axially symmetric 2D heat equation with Dirichlet and Neumann boundary conditions. Using cylindrical coordinate geometry, we describe a mathematical model of axially symmetric heat conduction for a stationary, homogeneous isotrophic solid with uniform thermal conductivity in a hollow cylinder with an exact solution in a particular case. We obtain the numerical solution of the PDE adapting finite difference and finite volume discretization techniques. Compared to the exact solution, we explore that the numerical schemes are the sufficient tools for the solution of linear or nonlinear PDE with prescribed boundary conditions. Furthermore, the numerical solution discrepancies in the results obtained from Explicit, Implicit and Crank-Nicolson schemes in Finite Difference Method (FDM) are extremely close to the exact solution in the case of Dirichlet boundary condition. The solution from the Explicit scheme is slightly far from the exactsolution and the solutions from Implicit and Crank-Nicolson schemes are extremely close to the exact solution in the case of Neumann boundary condition. Likewise, the numerical solutions obtained in the Finite volume method (FVM) are extremely close to the exact solution in the case of the Dirichlet boundary condition and slightly away from the exact solution in the case of the Neumann boundary condition.