1.1.9.1.2.1. Can be used on Unstructured Grids¶

There are no requirements on the grid to be structured (although structured grids can still be used). The Finite Volume Method is therefore suitable for complex geometries.

For 2D, Structured Grid means Quadrilateral i.e.

For 2D, Unstructured Grid means Triangle i.e.

• Computational efficiency Structured grids are less computationally efficient than unstructured grids, because of the rigid connectivity restriction. This resitrction may require high grid density where low grid density is needed - but this is dependent on the particular case.

• Memory requirements However, for the same number of grid points, structured grids require less memory to store than unstructured grids, because of the simpler connectivity.

• Convergence Structured grids may allow better convergence and higher resolution than unstructured grids and therefore higher accuracy, as the rigid connectivity limits the possibility for skewed elements.

• Geometry complexity Unstructured grids are more suited for highly complex geometries, whereas structured grids are better suited to simple geometries

Possiblities for the Finite Difference Method:

• The grid spacing can be non-uniform in the Finite Difference Method

• Local mesh adaption can be used with the Finite Difference Method.

• Curvilinear grids can be used, where the grid is transformed from cartesian to curvilinear and back again - cannot really have very complex geometries - the transformation must be smooth, but there is no such restruction on the Finite Volume Method. All the finite difference calculations are done on the cartesian grid and then the grid is transformed.

1.1.9.1.2.2. Uses Integral Formulation of Conservation Laws¶

• The native form of the conservation laws are integral

• We have set limits on the gradient by differentiating

• When we obtain the differential equations from the conservation laws, we are assuming that we can differentiate - i.e. that the solution is continuous

• If we have discontinuities, the integral formulation is more appropriate because we are not making the assumption of smoothness

1.1.9.1.2.3. Based on Cell Averaged Values¶

• Finite Difference approach uses points on a mesh

• Finite Volume Method uses cell-averaged values, areas in 2D, volumes in 3D, i.e.

• Discretisation form is a set of small cells - finite volumes

• Advantage: A conservative discretisation is automatically obtained, through the direct use of the integral conervation laws