Finite Volume Method: Literature Review

Finite volume method uses an integral approach to solve complex governing differential equations related to the problem. It uses gauss divergence theorem to convert volume integral to a surface integral

Need for gradient and interpolation schemes:

Gradient and interpolation schemes are needed when an estimate of an unknown field variable is required at a spatial point with the help for its consecutive nodes. Field values are known for adjacent nodes and used to estimate for the required nodes

Depending on various interpolation and gradient schemes, various estimates can be done depending upon the type of problem. For example, for pure conduction problem, the central differencing scheme can be used since it gives equal weightage to both of its adjacent node. If the problem is of a forced convection type, then the upwind scheme can be used where most of the weightage is given to the upstream node for a given finite volume

Similarly, QUICK scheme is also available where a quadratic profile of field variable is assumed with respect to the nodal position

Below image shows an example of a 1D heat diffusion equation, where FVM is used to discretize it, the final equation should be algebraically simplified, so gradient schemes and interpolation schems are used

Flux Limiters:

Flux limiter scheme is to limit the spatial derivatives to realistic values

The idea is to tune the numerical flux of high order and low order scheme using the flux limiter function in such a way that the resulting scheme gives a high order accuracy in the smooth region of flow and sticks with first order of accuracy of discontinuities

CONCLUSIONS:

In order to simplify a complex differential governing equation (the exact solution of which is difficult to obtain) related to a physical problem, FVM is used

To convert these equations further in linear algebraic forms, the gradient schemes and Interpolation schemes are applied