QUICK scheme

In computational fluid dynamics QUICK, which stands for Quadratic Upstream Interpolation for Convective Kinematics, is a higher-order differencing scheme that considers a three-point upstream weighted by quadratic interpolation for the cell face values.

In computational fluid dynamics there are many solution methods for solving the steady convection–diffusion equation.

[3] For the one-dimensional domain shown in the figure the Φ value at a control volume face is approximated using three-point quadratic function passing through the two bracketing or surrounding nodes and one other node on upstream side.

Let the two bracketing nodes be i and i − 1 and upstream node i – 2 then for a uniform grid the value of φ at the cell face between the three nodes is given by: The steady convection and diffusion of a property 'Ƥ' in a given one-dimensional flow field with velocity 'u' and in the absence of sources is given For the continuity of the flow it must also satisfy Discretizing the above equation to a control volume around a particular node we get Integrating this continuity equation over the control volume we get now assuming

The corresponding cell face values of the above variables are given by Assuming constant area over the entire control volume we get When the flow is in positive direction the values of the velocities will be

If Fw > 0 and Fe > 0 and if we use above equations for the convective terms and central differencing for the diffusion terms, the discretized form of the one-dimensional convection–diffusion transport equation will be written as: On re-arranging we get now it can be written in the standard form: where: When the flow is in negative direction the value of the velocities will be uw < 0 and ue < 0, For west face w the bracketing nodes are W and P, upstream node is E and for the east face E the bracketing nodes are P and E, upstream node is EE For

< 0 the flux across the west and east boundaries is given by the expressions : Substitution of these two formulae for the convective terms in the discretized convection-diffusion equation together with central differencing for the diffusion terms leads, after re-arrangement similar to positive direction as above, to the following coefficients.

In the QUICK scheme we face the problems of undershoot and overshoot due to which some errors occur.