FVM Interpolation and Gradient Schemes literature review

FVM:: Interpolation and Gradient Schemes literature review

Objective: Describe the need for interpolation schemes and flux limiters in FVM.

Need of Interpolation of schemes and Flux limiters in FVM

Interpolation of Schemes:

In the Finite volume method, the solution is available only at the computational nodes. But we need to calculate the value of variables that are not only present on the computational nodes, therefore we use interpolation techniques to calculate the values of a variable at non-computational nodes.

Types of interpolation schemes:

(1)Upwind Interpolation Scheme:

• In this scheme our solution never yields to the oscillatory solution.
• However it is only first-order accurate, hence we need very fine mesh.

The value of the variable at the node to be calculated is given by,

f(e)= f(P) ; if v > 0

= f(E) ; if v < 0

(2)Linear interpolation Scheme (CDS):

• The value at the non-computational node to be calculated is obtained by linear interpolation of two nearest computational nodes values.
• The linear interpolation is equivalent to the use of the central difference formula of the first order derivative and hence, this scheme is also termed as Central Differencing Scheme (CDS).
• This scheme is both first order as well as second-order accurate, in which the first-order accuracy for non uniform grid and second order accuracy for uniform grid.
• However, it is prone to oscillating solutions due to poor damping properties, when the grid spacing is large.

The value at the non-computational node to be calculated is obtained by,

f(e) = lambda(e)*F(E) + (1-lambda(e))*f(P)

where f(E) is the function value of node E

f(P) is the function value of node P

lambda(e) is the interpolation of the two computational node is given by,

lambda(e) = (x(e) – x(P) )/(x(e) – x(P))

(3)Quick Upwind Interpolation Scheme:

• This scheme is a combination of Upwind and CDS scheme, that means is has better damping properties than CDS scheme and better accuracy than Upwind scheme.

• The value at the non-computational nodes can be obtained by 3 nearest computational nodes, one downstream node D and two upstream nodes UU, is given by,

f(e) = f(U) + g1*(f(D) – f(U)) + g2*(f(UU) – f(U)

where f(U) is the function value of first upsteam node U

f(UU) is the function value of second upstream node UU

f(D) is the function value of downstream node D

g1 and g2 are the coefficients values which is given by,

g1 = ((x(e)-x(U)*(x(e)-x(UU))/((x(D)-x(U)*(x(D)-x(UU))

g2 = ((x(e)-x(U)*(x(e)-x(D))/((x(UU)-x(U)*(x(UU)-x(D))

Some other Schemes are,

Hybrid Interpolation Scheme: Blend of CDS and UDS schemebased on local Peclet number.

The value at the non-computational nodes can be obtained by,

f(e) = lambda* f(CDS) + (1-lambda)*f(UDS)

Total Variation Diminishing Scheme (TDS) :

The value at the non-computational nodes can be obtained by,

f(e) =f(P) + 0.5*psy(r)*(f(E) – f(P))

Need of Flux Limiter: For a convective flow problem, the use of Interpolation schemes gives us a good solution. But for shock tube problems these schemes are not suitable in that case we need flux limiters to restrict long jumping of solution through discontinuities and to ascertain our numerical values are close to analytical values.