Week 9 - FVM Literature Review

Aim: Literature review of Finite volume method and also describe the need for interpolation scheme and flux limiters in FVM.

Finite Volume Method:

The most widely applied method today in CFD is the FVM, which discretizes directly the integral form of the conservation laws.

Its popularity is due to the its generality, its conceptual simplicity and the relative ease off application to both structured as well as to any kind of unstructured grids.

The finite volume method (FVM) is a method for representing and evaluating partial differential equations in the form of algebraic equations. In the finite volume method, volume integrals in a partial differential equation that contain a divergence term are converted to surface integrals, using the divergence theorem.

These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative. Another advantage of the finite volume method is that it is easily formulated to allow for unstructured meshes. The method is used in many CFD packages. "Finite volume" refers to the small volume surrounding each node point on a mesh.

Difference between FDM and FVM:

Example: 1D heat conduction Equation

Interpolation schemes and Flux limiters in FVM:

– First order upwind scheme.
– Central differencing scheme.
– Power law scheme.
– Second order upwind scheme.
– QUICK scheme.

Flux limiters:

Flux limiters are used in high resolution schemes – numerical schemes used to solve problems in science and engineering, particularly fluid dynamics, described by partial differential equations (PDE's).

Most of us might have seen the behaviour of numerical schemes in order to capture shocks and discontinuity that arises in hyperbolic equations. Physically, these equations model the convective fluid flow. It has been observed that low-order schemes are usually stable but quite dissipative in nature around the points of discontinuity/shocks. On the other hand higher-order numerical schemes are unstable in nature and show oscillations in the vicinity of discontinuity. One can have a better understanding of such behaviour by analysing the modified equation of these schemes.

Now as said above one can not have high order accuracy without oscillations. The objective is to have highly accurate and stable oscillation free schemes. This kind of schemes are known as high resolution schemes. In 1984 Harten proposed the constuction and gave TVD criteria for such scheme. Here we talk only about using flux limiters function to construct high resolution schemes. 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 in the vicinity of socks/discontinuities as follows:

Define the numerical flux fuction of high resolution conservative scheme as

where are the numerical flux of conservative low order and high order schemes respectively. and is a function of smoothness parameter usually defined as

What remains next is to define the limiter fuction in such a way that it satisfies at least the following properties:

• remains positive ,
• satisfies
• passes through a perticular region known as TVD region associted with the underlying scheme in order to guarantee stability of the scheme.