Week 9 - FVM Literature Review

The Finite Volume Method (FVM) is a numerical technique used for solving partial differential equations (PDEs) by discretizing the domain into small control volumes or cells. These control volumes are used to integrate the governing equations over the domain, converting them into algebraic equations that can be solved iteratively.

Major differences between Finite Difference Method (FDM) and Finite Volume Method (FVM):

1. Discretization approach:

- FDM discretizes the domain by approximating derivatives at discrete points (nodes), while FVM discretizes the domain by integrating the governing equations over control volumes.

2. Conservation form:

- FVM naturally conserves quantities such as mass, momentum, and energy, making it well-suited for problems governed by conservation laws. FDM may require additional manipulation to ensure conservation.

3. Control volumes:

- FVM treats the physical domain as a collection of control volumes, with fluxes across the faces of these volumes representing the transport of quantities. FDM typically does not utilize control volumes explicitly.

4. Stencil size:

- FDM often uses symmetric stencils to approximate derivatives, resulting in uniform grid spacing. FVM allows for irregular grid spacing and can handle non-uniform grids more naturally.

Interpolation schemes and flux limiters are essential components of FVM, especially when dealing with discontinuous solutions or sharp gradients. In the context of the 1D linear heat conduction equation:

dT/dt = a.dT2/d2x

where T �T is temperature, t �tt is time, x� is spatial coordinate, and a i�aas thermal diffusivity.

Interpolation schemes are used to estimate the values of temperature at cell faces, as the temperature distribution within a cell is typically represented by a piecewise constant or linear function. Common interpolation schemes include linear interpolation and piecewise constant interpolation.

Flux limiters are employed to prevent spurious oscillations and maintain numerical stability, particularly when solving hyperbolic conservation laws or when dealing with discontinuous solutions. Flux limiters restrict the magnitude of the fluxes to ensure that they do not exceed physical bounds, effectively dampening oscillations and maintaining the accuracy of the solution. Common flux limiters include the minmod limiter, superbee limiter, and van Leer limiter.