FVM Interpolation and Gradient Schemes literature review
Methods of discretization:
Most of the physical phenomena inherently can be described by a partial differential equation governing the change of a particular variable over spatial and time dimensions. These equations describing variables are valid over a particular domain and need to be discretized for better accuracy. This can be described by either
1)Finite difference method: here domain discretized into grids and variable is defined at grid points and derivatives are defined in equivalent form by Taylor series expansion. these equations are solved at grid points and thus gives us the entire simulation.
2) Finite volume method: here entire volume is discretized into smaller volumes and the variable is averaged over that control volume center point. This gives better control over local and global variations. This volume cell has six faces and over which integrals need to be approximated which is done by interpolation schemes like upwind interpolation scheme, linear interpolation scheme, and quadratic interpolation scheme, etc.
3) Finite element method: here value is calculated at grid points and then interpolated by shape function to get a better picture of simulation.
4) Interpolation schemes in Finite volume method:
1) Quick interpolation scheme: In order to find the cell face value, a quadratic function passing through two bracketing or surrounding nodes and one node on the upstream side must be used.
2)Upwind scheme: Consider a typical grid point i in the domain. In a one-dimensional domain, there are only two directions associated with a point i – left (towards negative infinity) and right (towards positive infinity). If u is positive and the traveling wave solution of the equation above propagates towards the right, the left side of i is called upwind side and the right side is the downwind side. Similarly, if u is negative the traveling wave solution propagates towards the left, the left side is called downwind side and the right side is the upwind side.
In linear interpolation scheme is second-order accurate and upwind scheme is the first order accurate as that derivative is included and the second-order derivative is ignored whereas QUICK does take the second-order derivative into account, but ignores the third-order derivative hence this is considered third order accurate.
5) Flux limiters:
Flux limiters are used in numerical schemes to solve problems in science and engineering, particularly fluid dynamics, that are described by (PDEs). Their main purpose is to avoid the spurious oscillations (wiggles) that would otherwise occur with high-order spatial discretization due to shocks, discontinuities, or steep gradients in the solution domain.
How They Work
The main idea behind the construction of flux limiter schemes is to limit the spatial derivatives to realistic values—for scientific and engineering problems, this usually means physically realizable values. They are used in high-resolution schemes for solving problems described by PDE\'s and only come into operation when sharp wavefront is present. For smoothly changing waves, the flux limiters do not operate and the spatial derivatives can be represented by higher-order approximations without introducing nonreal oscillations.
dT/dt=(alpha)*(d^2T/dx^2)
where, for a finite volume scheme, Fui+12\">F(T(i+1/2)) and Fui−12\">F(T(i−1/2)) represent temperature edge fluxes for the ith cell. If these fluxes can be represented by low- and high- resolution schemes, then a flux limiter can switch between these schemes depending upon the gradients close to the particular cell as followsF(T(i+1/2))=f(i+1/2)low−ϕ(ri)*[f(i+1/2)low−f(i+1/2)high]
F(T(i-1/2))=f(i-1/2)low−ϕ(ri)*[f(i-1/2)low−f(i-1/2)high]
where, flow = low-resolution flux, fhigh = high-resolution flux, ϕ(r) = flux limiter function, and r represents the ratio of successive gradients on the solution mesh, i.e.,
The limiter function is constrained to be greater than or equal to zero, i.e., r ≥ 0. Therefore, when the limiter is equal to zero (sharp gradient, opposite slopes, or zero gradients), 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. The various limiters listed below have differing switching characteristics and are selected to suit the particular problem and numerical solution scheme. No particular limiter has been found to work well for all problems, and a particular choice is usually made on a trial-and-error basis.