# Approximation Schemes for convective term - structured grids - Summary of Discretizations Schemes and examples

When we shall fill this page, I offer to make common identifications, because in different issues was used different notation.

Also we beg everybody to help me with original works. Later I shall write, what is necessary. If anyone have literature connected with convective schemes, please drop me a line.

We shall be very glad and grateful to hear any critical suggestion (please drop a few lines at Wiki Forum)

It is just a skeleton, but we hope that it will be developed into the good thing

## Numerical examples

### Pure convection of a scalar step by a rotating velocity field (Smith-Hutton problem)

R.M.Smith and A.G.Hutton (1982), "The numerical treatment of advection: A performance comparison of current methods", Numerical Heat Transfer, Vol. 5, p439.

This was the test problem devised for evaluating a range of numerical models of convection at the third meeting of the International Association for Hydraulic Research Working Group on Refined Modelling of flow

Sometimes it was used scalar profile with a discontinuity at $\boldsymbol{x=-0.5}$

We shall use here more smooth inlet profile

 $la-la$ (1)

provides a simple problem with a strong discontinuity in a scalar profile and flow that is not parallel to the boundaries of the domain being tested. As such it should reveal the poor convergence of the first order schemes, which exhibit false diffusion on flow that is not parallel to the grid, whilst the sharp gradient should generate oscillations in the solutions generated using the second and third order schemes.

The steady transport equation is solved in the region $-1\leq x \leq 1$ \ , \ $0 \leq y \leq 1$, with the streamfunction being specified as

 $\psi = - \left( 1 - x^{2} \right) \left( 1 - y^{2} \right)$ (1)

which is shown in figure below. This streamfunction gives a velocity field of

 $u = 2y \left( 1 - x^{2} \right)$ (1)
 $v = -2x \left( 1 - y^{2} \right)$ (1)

## Example code for solving Smith-Hutton problem

Dear friends

It's just a scrap. Later I'll correct it, although it's a complete working code

Michail

It's a results, obtained using this code (UDS and HLPA schemes)

Below it's cleary seen the numerical diffusion impact, comparing the contour fields obtaining using the UDS and HLPA. A bit later we shall place here a solution gained with QUICK scheme, and it will be seen the osscilations.