Week 9 - FVM Literature Review

Finite Volume Method (FVM) -

The Finite Volume Method (FVM) is a discretization method for the approximation of a single or a system of partial differential equations expressing the conservation, or balance, of one or more quantities. These partial differential equations (PDEs) are often called conservation laws; they may be of different nature, e.g. elliptic, parabolic or hyperbolic, and they are used as models in a wide number of fields, including physics, biophysics, chemistry, image processing, finance, dynamic reliability. They describe the relations between partial derivatives of unknown fields such as temperature, concentration, pressure, molar fraction, density of electrons or probability density function, with respect to variables within the domain under consideration.

Finite Difference Method (FDM) -

The finite difference method (FDM) is an approximate method for solving partial differential equations. It has been used to solve a wide range of problems. These include linear and non-linear, time independent and dependent problems. This method can be applied to problems with different boundary shapes, different kinds of boundary conditions, and for a region containing a number of different materials.

Major differences between FDM & FVM -

The main difference between the FVM and FDM method is its high flexibility in the implementation of the discretization method. In the FVM, the discretization process is carried out directly in the physical domain without any transformation between the computational and physical coordinate systems. In addition, FVM is a strictly conservative method. On other hand FDM use partial differential equations. FDMs are usually used for regular boundary domains, and equally spaced meshes.

Interpolation schemes -

Interpolation technique is used where approximation of surface and volume integral require the value of variable at location than computaional nodes of the control volume.

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 (PDEs). They are used in high resolution schemes, such as the MUSCL scheme, to avoid the spurious oscillations (wiggles) that would otherwise occur with high order spatial discretization schemes due to shocks, discontinuities or sharp changes in the solution domain. Use of flux limiters, together with an appropriate high resolution scheme, make the solutions total variation diminishing (TVD).