FVM Flux Limiters and Interpolation Schemes

FDM vs FVM: In, finite difference method the computational domain is discretized into points and the equations are converted into discretized equations using various difference schemes (central difference scheme, forward difference scheme, backward difference scheme).

In order to get the solution, the discretized equation is then solved for individual points of the computational domain until the convergence is obtained.

Above figure shows discretisation of the domain in points or nodes in the finite difference method.

Above figure shows discretisation of the domain in small control volumes in the finite volume method.
However, in a finite volume method, the computational domain is divided into arbitrarily shaped control volumes and the solution is solved for the same. Since the governing equations are valid at one control volume hence it is valid at all control volumes of the computational domain. This property helps us to compute the solution of the whole domain irrespective of the shape and size of the domain. Finite volume refers to the small volume around each node point in the mesh. The volume integrals in the partial differential equation that contain divergence term are converted to surface integrals. Using these terms, fluxes at each surface are evaluated. Since the flux entering each control volume is constant, this method becomes conservative in nature. In this way, the properties are conserved in FVM.

Consider the figure above. When evaluating the values at point P, the values at the surfaces `e` and `w` are unknown. Hence, a method of mathematical approximation is required to compute the values at these surfaces. These mathematical approximations are known as interpolation schemes.

Flux limiters: General lower-order schemes are often not fit for accurate resolution of steep gradient phenomena as they generally induce non-physical effects like oscillations or wiggles where shocks or discontinuities are present. To overcome this problem flux limiters are used in high-resolution methods. In high-resolution methods, the flux limiters limit the solution gradient near the shocks instead of allowing it to grow which would have failed to reach a solution. They are also called as slope limiters as they both have same mathematical form. The aim of flux limiter is to limit the spatial derivatives to realistic or physical values.

Consider the figure above where horizontal lines represent cell volume average value of that cell for a given flow field `u`.

Without flux limiter - If the successive calculation of values in next time step is calculated using cell centered average and without a flux limiter then the next value obtained may be greater than the local extremum of the successive cells. Hence, the difference between analytical solution and computed solution will increase in sucessive cell calculations(as shown in red line). Hence, the non physical effects like oscillations gets introduced.

With flux limiter - The limiter function is constrained to be greater than or equal to zero. Therefore, when the limiter is equal to zero (sharp gradient, opposite slopes or zero gradient), the flux is represented by a low resolution scheme. Similarly, when the limiter is equal to 1 (smooth solution), it is represented by a high resolution scheme. Hence,the solution doesn\'t blow up and attains convergence in successive calculations(as shown in green line).

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 the first order of accuracy in the vicinity of shocks/discontinuities. Hence, the flux limiter acts as a switch and helps in maintaining the slope of physical values in the vicinity of shocks/discontinuities.