HR scheme = Higher Order Scheme + a Convection Boundedness Criterion

HR scheme can be obtained by either of the following method

- Smoothening an unbounded Higher Order scheme with diffusive flux.
- Enhancing a first order upwind scheme with a limited anti-diffusive flux (TVD schemes).

Two frameworks used for convection boundedness are

- Normalized Variable Formulation (NVF)
- Total Variation Diminishing (TVD) (Enhancing a first order upwind scheme with a limited anti-diffusive flux)

**Normalized Variable Formulation (NVF)**

In this method, the dependent variable is normalised locally using the upwind ( \phi_{C}), downwind ( \phi_{D}) and far upwind ( \phi_{U}) values to get the value at the face ( \phi_{f})

The normalised value is defined as \tilde{\phi} = (\phi - \phi_{U})/ (\phi_{D} - \phi_{U})

The normalised value of \phi_{U} and \phi_{D} becomes 0 and 1 respectively and \tilde{\phi_{f}} = f(\tilde{\phi_{C}})

For example a QUICK Scheme which is defined for a structured grid changes as follows:

\phi_{f} = \frac{3}{8} \phi_{D} + \frac{3}{4}\phi_{C} -\frac{1}{8} \phi_{U} ====> \tilde{\phi_{f}} = \frac{3}{8} + \frac{3}{4} \tilde{\phi_{C}}

**The Convection Boundedness Criterion**

The normalised face value \tilde{\phi_{f}} shall be bounded as follows

min(\tilde{\phi_{C}},1) \le \tilde{\phi_{f}} \le max(\tilde{\phi_{C}},1)

and should pass through the points (0,0) and (1,1) when plotted \tilde{\phi_{C}} versus \tilde{\phi_{f}}. If \tilde{\phi_{C}} is less than 0 or greater than 1, \tilde{\phi_{f}} assumes the value of \tilde{\phi_{C}} (upwind). This ensures damping out the extrema. This is explained in the page 438 of ^{1}

**Constructing a High Resolution Scheme**

A HR scheme can be made while following the guidline of \tilde{\phi_{f}} passing through (0,0), (1,1) and assuming upwind values outside the range of 0 and 1. All second order schemes has to pass through (1/2, 3/4), to have second order accuracy. In addition to that, if the slop at this point is 3/4, then the scheme will be third order accurate. Hence almost all schemes are made to pass through this point. A SMART scheme derived from QUICK is implemented as follows:

\tilde{\phi_{f}} = \begin{cases} \frac{3}{4}\tilde{\phi_{c}}+\frac{3}{8} &\text{if } & 0\le \tilde{\phi_{c}} \le \frac{5}{6}\\ 1 &\text{if } & \frac{5}{6} \le \tilde{\phi_{c}} \le 1\\ \tilde{\phi_{c}} &\text{if } & \text{otherwise} \end{cases}

Between 5/6 and 1 the value of \tilde{\phi_{f}} is constant. Convergence can be improved if such a constant line can be avoided. Hence a modified scheme was proposed as

\tilde{\phi_{f}} = \begin{cases} 3\tilde{\phi_{c}} &\text{if } & 0\le \tilde{\phi_{c}} \le \frac{1}{6}\\ \frac{3}{4}\tilde{\phi_{c}}+\frac{3}{8} &\text{if } & \frac{1}{6}\le \tilde{\phi_{c}} \le \frac{7}{10}\\ \frac{1}{3}\tilde{\phi_{c}}+\frac{2}{3} &\text{if } & \frac{7}{10} \le \tilde{\phi_{c}} \le 1\\ \tilde{\phi_{c}} &\text{if } & \text{otherwise} \end{cases}

2. **Total Variation Diminishing (TVD)**

Total variation is defined as the sum of absolute difference between the consecutive nodal values. i.e., sum(abs( \phi_{i+1} - \phi_{i} )). If this is not increasing as the solution progresses, the method could be termed TVD.

The upwind method is very stable but excessively diffusive. Central Difference scheme is having no numerical diffusion but has dispersion errors. An intelligent combination is sought in such a way that a portion of anti-diffusive flux of central difference scheme (1/2 * ( \phi_{D} - \phi_{C} )) is added to upwind scheme (\phi_{f} = \phi_{C}). For that a limiter is introduced wherever large gradients are present.

Limiter is represented as \psi(r) where r is the ratio of gradients at nodes across the face in consideration.

\phi_{f} = \phi_{C} + \frac{1}{2}\psi(r_{f})(\phi_{D} - \phi_{C} ) \text{ where } r_{f} = \frac{\phi_{C}-\phi_{U}}{\phi_{D}-\phi_{C}}

For unsteady 1D convection

\frac{\partial(\rho \phi)}{\partial t} = - \frac{\partial(\rho u \phi)}{\partial x}

where the right hand side can be written as -a(\phi_{C}-\phi_{U}) + b(\phi_{D}-\phi_{C}) , the condition for monotone scheme is

a\ge0 , b\ge0 \text{ and } 0 \le a+b\le 1

which gives a constraint for \psi(r_{f}) as follows:

\psi_{f}(r_{f}) = \begin{cases} min(2r_{f},0) & r_{f}\ge 0\\ 0 & r_{f}\le 0 \end{cases}

for central difference scheme \phi_{f} is defined as

\phi_{f} = \phi_{C} + \frac{1}{2}(\phi_{D} - \phi_{C} )

Hence \psi(r_{f}) = 1 for the case of central difference scheme. For second order upwind scheme, \phi_{f} is defined as

\phi_{f} = \frac{3}{2}\phi_{C} - \frac{1}{2}\phi_{U}

Hence by comparing, \psi(r_{f}) for second order upwind method is obtained as

\psi(r_{f}) = \frac{\phi_{C}-\phi_{U}}{\phi_{D}-\phi_{C}} \text{ which is } r_{f} \text{ itself }

\text{ for QUICK }\psi(r_{f}) = \frac{ 3 + r_{f} }{4}