A new numerical approach to solve an elliptic equation

A new numerical approach to solve an elliptic partial differential equation that originates from the governing equations of steady state fluid flow and heat transfer is presented. The elliptic partial differential equation is transformed by introducing an exponential function to eliminate the convection terms in the equation. A fourth-order central differencing scheme and a second-order central differencing scheme are used to numerically solve the transformed elliptic partial differential equation. Analytical solutions of this equation are also given. Comparisons are made between the analytical solutions, the numerical results using the present schemes, and those using the four classical differencing schemes, namely, the first-order upwind scheme, hybrid scheme, power-law scheme, and exponential scheme. The comparisons illustrate that the proposed algorithm performs better than the four classical differencing schemes.