Week 9 - FVM Literature Review

FVM LITERATURE REVIEW

The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In finite volume method, the volume integrals in the partial differential equation containing a divergence term are converted into surface integrals using the divergence theorem.These terms are then evaluated as fluxes at the surfaces of each finite volume. Since, the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. These methods are easy to formulate to allow for unstructured meshes. Hence, finite volume refers to the small volume surrounding each node point on a mesh.

DIFFERENCE BETWEEN FVM AND FDM

1.Form of discretisation based on the integral form of conservation equation.

2.In FDM the computational domain is represemted as set of parts where as finite volume method deals with a small volume(finite volume) within which the governing equations are solved and integrated throughout the domain.

3.In FDM method the equations are solved at a point, in contrast the equations are solved for a particular volume in FVM method.

4.In FVM method the conservative properties are automatically included in the equation where as in FDM they are not included.

5.The FVM methods can be compared and contrasted with FDM methods, which approximates derivative using nodal values, FDM on the other hand, creates local approximations of the solution using local data and constructs a global approximation by combining them together.

INTERPOLATION

It is a process of calculating the unknown values by using the known values.In case of FVM methods the approximation of surface and volume integrals require the values of variables at a location other than computational points of the control volume, hence, FVM methods require various interpolation schemes such as,

• First order UPWIND scheme which requires the definition of only one upstream value of all cell centres.
• Second order UPWIND scheme where two values from the upstream of all cell centres need to be defined.
• Quadratic UPWIND interpolation scheme which involves two upstream and one downstream value from all cell centres.
• Apart from the above methods, forward difference scheme, backward difference scheme and central differencing scheme are also used.

FLUX LIMITERS

• Flux limiters are used in high resolution schemes, particularly in fluid dynamics as they deal with partial differential eaquations(PDE's).
• They are used to avoide wiggles that would otherwise occur with high order spatial discretisation schemes due to shocks, discontinuities or sharp changes in the solution domain.
• Use of flux limiters with appropriate high resolution schemes, make solutions total variation diminishing (TVD).

NEED FOR FLUX LIMITERS

• Interpolation schemes provide reliable reults but in case of problems such as shock flow problems due to discontinuities, there arises a need for flux limiters.
• The limiter function can be constrained to greater than pr equal to zero, there by representing a low resolution scheme.
• In contras, when the limiter is equal to 1 it is represented by a high resolution scheme.