Week 9 - FVM Literature Review

The Finite Volume Method is a method used to evaluvate partial differential equations by converting them into algebaric equations. In this method, each domain is divided into finitely small volumes, the partial differential equations are solved for each volume. Because solving for each volume will be tedious, the equations are evaluvated to a volumetric or surface integral.

Upon calculating the integral as mentioned above, one can notice the equations are basically calculating the flux-es. Hence, the equations are conserved. The volumes which are divided from the domain can be of any shape, such as a cubiod, cube or tetrahedron.

Consider this 1D steady state diffusion problem's governing equation:

across this domain

When this equation is solved over a volumetric intergral we get,

This equation is essentially calculating difference in flux-es entering and exiting the domain.

Differences between FVM and FDM

FVM

1. FVM can be solved for any irregularly shaped domain, as the domain is divided into volumes of any shape and size.

2. FVM uses integral forms of partial differential equations to solve in each control volume.

3. It is also easier to understand but harder to implement by simple arithmetic operations in computer programming.

4. Governing equations can only be solved conservatively.

FDM

1. FDM can only be solved for regular/straight objects, since the entire domain is divided to nodes spaced at equal intervals.

2. FDM uses Taylor Series' approximations to solve the partial differential equations between nodes in the domain.

3. It is easier to understand, and easier for a human to code as it involves basic interpolation concepts that can be done using simple arithmetic equations.

4. Governing equations can be solved by both conservative and non-conservative forms.

Interpolation Methods and their need

In FVM, since the solution only calculates the values of any variable 1 point of each finite volume (usually center of volume), there needs to be well defined interpolation methods to find interpolated values accurately.

- Upwind Interpolation (UDS): Approximation is done by using the upstream value of the variable. Similar to forward differencing in FDM

- Linear Interpolation (CDS): Finds the linear interpolation between the 2 nearest nodes using an interpolation factor.

- Quadratic Upwind Interpolation (QUICK): approximate profile using quadratic profiles and a 3rd node.

- Higher order schemes: These are required if the integrals are also higher order approximations.

Flux limiters and their need

When solving lower resolution solutions (CFL number is near 1), there are less discontinuities in the solution this could be because of the lower order solving methods used or other reasons. But when higher resolution schemes are used, a lot of discontinuities arise. These could be due to sharp changes in the domain, very small mesh sizes, low CFL number, etc. This discontinuity will cause the solution to slowly turn unstable and ruin the solution. To prevent this happening, a Flux limiter is used to avoid these oscillations. In simple words, they are fucntions used to limit the values of variables to realistic values (less than 1e6 or greater than 1e-5, etc.)

Sources:

- https://ocw.mit.edu/courses/mechanical-engineering/2-29-numerical-fluid-mechanics-spring-2015/lecture-notes-and-references/MIT2_29S15_Lecture16.pdf

- https://en.wikipedia.org/wiki/Flux_limiter

- https://folk.ntnu.no/leifh/teaching/tkt4140/._main074.html