Finite Volume Method for Solving 1D and 2D Convection-Diffusion Problems

Finite Volume Method (FVM) is a numerical technique for solving partial differential equations, particularly well-suited for one-dimensional and two-dimensional convection-diffusion problems. In implementation, FVM involves dividing the domain into discrete control volumes and applying conservation laws to each volume. Common discretization schemes include central differencing and power-law scheme. Central differencing is a straightforward approach that approximates derivatives using values from adjacent grid points - for instance, the convective term might be implemented as (φ_E - φ_W)/2Δx where φ represents the transported quantity. While simple to code, this method can produce unphysical oscillations at high Peclet numbers. The power-law scheme offers a more sophisticated alternative, where terms are raised to specific powers to achieve higher accuracy. This scheme can be implemented using exponential functions or approximated with polynomial expressions that provide better stability characteristics. In code implementation, this typically involves conditional statements based on local Peclet number calculations. Both methods present distinct advantages and limitations: central differencing is computationally efficient but may require additional stabilization, while power-law scheme provides better physical accuracy at the cost of increased complexity. The choice between them depends on specific problem requirements, computational resources, and desired accuracy levels.