- Practice · 4 marks
Compare FDM, FVM and FEM
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.
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?
| 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
- It is derived from the integral form, so mass, momentum and energy are conserved locally in each control volume, even on a coarse mesh.
- It works on unstructured meshes, so complex industrial geometry can be meshed automatically.
- The physical meaning of each term (flux through a face) is clear, which makes boundary conditions and upwind schemes natural.
- 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.
✓ Answer checked by Claude Opus 5.5 (high effort)
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