--- title: "Compare FDM, FVM and FEM" subject: "Applied Computational Fluid Dynamics" chapter: 3 key: "semester_7/applied_computational_fluid_dynamics:fdm-fvm-fem-comparison" source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/fdm-fvm-fem-comparison site: BCT Engineering --- # Compare FDM, FVM and FEM Applied Computational Fluid Dynamics · Chapter 3: Discretization Methods - Asked in: year not given, 4 marks - Video: [Difference between Finite Difference Method, Finite Volume Method and Finite Element Method](https://www.youtube.com/watch?v=DcZP9pOK66Y) — Degree of Freedom - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/fdm-fvm-fem-comparison - Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-3 ## Question Differentiate between the finite difference, finite volume and finite element methods used in CFD. Why is the finite volume method preferred in most commercial CFD codes? ## Short answer FDM uses Taylor series at grid points, simple but structured meshes, conservation not guaranteed. FVM integrates conservation laws over control volumes, ensures local conservation, handles unstructured meshes, fluxes natural for boundary conditions. FEM uses weighted residuals with shape functions, global conservation, complex geometry. FVM preferred in commercial CFD for local conservation, unstructured meshes, physical flux interpretation, and efficiency with pressure-based solvers like SIMPLE. ## Answer | Point | Finite difference (FDM) | Finite volume (FVM) | Finite element (FEM) | |---|---|---|---| | Basis | Taylor series; PDE at grid points | Integral conservation law over each cell | Weighted residual / variational form with shape functions | | Unknowns | Point values | Cell-average values | Nodal values with interpolation inside the element | | Mesh | Mainly structured | Structured or unstructured | Unstructured, complex shapes | | Conservation | Not guaranteed | Exact (flux in = flux out for each cell) | Global, not local, unless special form | | Complex geometry | Difficult | Easy | Very easy | | Typical use | Research, simple geometry, heat conduction | Fluent, OpenFOAM, STAR-CCM+, CFX | Structural analysis, some fluid problems | | Simplicity | Simplest to code | Moderate | More complex mathematically | #### Why FVM is preferred 1. It is derived from the integral form, so mass, momentum and energy are **conserved locally** in each control volume, even on a coarse mesh. 2. It works on unstructured meshes, so complex industrial geometry can be meshed automatically. 3. The physical meaning of each term (flux through a face) is clear, which makes boundary conditions and upwind schemes natural. 4. It is efficient in memory and suited to segregated pressure-based solvers such as SIMPLE. FDM remains useful for teaching, simple domains and high-order schemes, because its formulae follow directly from the Taylor series. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which pairing of method and basis is correct? A. FDM: integral conservation over cells; FVM: Taylor series B. FDM: Taylor series at grid points; FVM: integral conservation over each cell C. FDM: shape functions; FEM: Taylor series D. FVM: weighted residual with shape functions; FEM: cell fluxes Answer: B. FDM approximates the PDE at points using Taylor series, while FVM applies the integral conservation law over each control volume. (Key point: Basis of each method) 2. Why does FVM conserve mass, momentum and energy locally even on a coarse mesh? A. It uses higher-order Taylor series B. It always uses structured meshes C. It balances the flux in and flux out of each control volume from the integral form D. It uses nodal shape functions Answer: C. Because each cell's balance is written in integral form, the flux leaving one cell is exactly the flux entering its neighbour. (Key point: Local conservation) 3. Which is NOT given as a reason commercial CFD codes prefer FVM? A. It works on unstructured meshes for complex geometry B. Face-flux terms have clear physical meaning, making upwind schemes and boundary conditions natural C. It suits segregated pressure-based solvers like SIMPLE D. It is the simplest method to code, simpler than FDM Answer: D. FDM is the simplest to code; FVM is of moderate difficulty. Its advantages are conservation, unstructured meshes, physical flux meaning and solver suitability. 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