Approximation Schemes for convective term

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1 Discretisation Schemes for convective terms in General Transport Equation. Finite-Volume Formulation, structured grids
2 Introduction
3 Basic Equations of CFD
4 Convection Schemes
5 Basic Discretisation schemes
6 High Resolution Schemes (HRS)
- 6.1 Classification of High Resolution Schemes
  - 6.1.1 Linear schemes
  - 6.1.2 Non-Linear schemes
- 6.2 Numerical Implementation of HRS
7 Normalised Variables Formulation (NVF)
8 Normalised Variable and Space Formulation (NVSF)
9 Normalised Variables Diagram (NVD)
10 Total Variation Diminishing (TVD)
11 Total Variation Diminishing Diagram (Sweby diagram)
12 Convection Boundedness Criterion (CBC)
13 Schemes
14 Summary of Discretizations Schemes
- 14.1 Discretizations Schemes Estimation of order
- 14.2 Selection advice
15 Comparison of Discretizations Schemes
16 Numerical examples
- 16.1 Pure convection of a scalar step by a rotating velocity field (Smith-Hutton test)
- 16.2 Square Lid-driven cavity flow
17 Example code for solving Smith-Hutton test

Discretisation Schemes for convective terms in General Transport Equation. Finite-Volume Formulation, structured grids

Introduction

Here is described the discretization schemes of the convective terms in the finite-volume equations. The accuracy, numerical stability and the boundness of the solution depends on the numerical scheme used for these terms. The central issue is the specification of an appropriate relationship between the convected variable, stored at the cell centre and its value at each of the cell faces.

Basic Equations of CFD

All the conservation equations can be written in the same generic differential form:

$\frac {\partial( \rho \phi )} {\partial t} + \frac{\partial}{\partial x_{i}} \left( \rho U \phi - \Gamma_{\phi} \frac{\partial\phi}{\partial x_{i}}\right)=S_{\phi}$

(1)

Equation (1) is integrated over a control volume and the following discretised equation for $\boldsymbol{\phi}$ is produced:

$\boldsymbol{ J_{h}- J_{l} + J_{n}- J_{s} + J_{e}- J_{w} + D_{h} - D_{l} + D_{n} - D_{s} + D_{e} - D_{s} = S_{p}}$

(2)

where $\boldsymbol{S_{p}}$ is the source term for the control volume $\boldsymbol{P}$ , and $\boldsymbol{J_{f}}$ and $\boldsymbol{D_{f}}$ represent, respectively, the convective and diffusive fluxes of $\boldsymbol{\phi}$ across the control-volume face $\boldsymbol{f}$ $\boldsymbol{(f=h,l,n,s,e,w)}$

The convective fluxes through the cell faces are calculated as:

$\boldsymbol{ J_{f}=C_{f}\phi_{f} }$

(1)

where $C_{f}$ is the mass flow rate across the cell face $f$ . The convected variable $\phi_{f}$ associated with this mass flow rate is usually stored at the cell centres, and thus some form of interpolation assumption must be made in order to determine its value at each cell face. The interpolation procedure employed for this operation is the subject of the various schemes proposed in the literature and the accuracy, stability and boundedness of the solution depends on the procedure used.

In general, the value of $\boldsymbol{\phi_{f}}$ can be explicity formulated in terms of its neighbouring nodal values by a functional relationship of the form:

$\phi_{f}=P \left( \phi_{nb} \right)$

(1)

where $\boldsymbol{\phi_{nb}}$ denotes the neighbouring-node $\boldsymbol{\phi}$ values. Combining equations (\ref{eq3}) through (\ref{eq4a}), the discretised equation becomes:

$\left\{ D_{h} + C_{h} \left[ P \left( \phi_{nb} \right) \right]_{h} \right\} - \left\{ D_{l} + C_{l} \left[ P \left( \phi_{nb} \right) \right]_{l} \right\} + \left\{ D_{n} + C_{n} \left[ P \left( \phi_{nb} \right) \right]_{n} \right\} - \left\{ D_{s} + C_{s} \left[ P \left( \phi_{nb} \right) \right]_{s} \right\} + \left\{ D_{e} + C_{e} \left[ P \left( \phi_{nb} \right) \right]_{e} \right\} - \left\{ D_{w} + C_{w} \left[ P \left( \phi_{nb} \right) \right]_{w} \right\} = S_{p}$

(1)

Convection Schemes

All the convection schemes involve a stencil of cells in which the values of $\boldsymbol{\phi}$ will be used to construct the face value $\boldsymbol{\phi_{f}}$

Where flow is from left to right, and $\boldsymbol{f}$ is the face in question.

$\boldsymbol{u}$ - mean Upstream node

$\boldsymbol{c}$ - mean Central node

$\boldsymbol{d}$ - mean Downstream node

.

Basic Discretisation schemes

Central Differencing Scheme (CDS)

The most natural assumption for the cell-face value of the convected variable $\boldsymbol{\phi_{f}}$ would appear to be the CDS, which calculates the cell-face value from:

$\phi_{f}=0.5 \left( \phi_{c} + \phi_{d} \right)$

(1)

This scheme is 2nd-order accurate, but is unbounded so that unphysical oscillations appear in regions of strong convection and also in the presence of discontinuities such as shocks. The CDS may be used directly in very low Reynolds-number flows where diffusive effects dominate over convection.

Upwind Differencing Scheme (UDS)

The UDS assumes that the convected variable at the cell fase $\boldsymbol{f}$ is the same as the upwind cell-centre value:

$\boldsymbol{\phi_{f}= \phi_{c} }$

(1)

The UDS is unconditionally bounded and highly stable, but as noted earlier it is only 1st-order accurate in terms of truncation error and may produce severe numerical diffusion. The scheme is therefore highly diffusive when the flow direction is skewed relative to the grid lines.

Hybrid Differencing Scheme (HDS)

The HDS of Spalding [1972] switches the discretisation of the convection terms between CDS and UDS according to the local cell Peclet number as follows:

$\phi_{f}=0.5 \left( \phi_{c} + \phi_{d} \right) \mbox{ for } Pe \triangleleft 2$

(1)

$\phi_{f}= \phi_{c} \mbox{ for } Pe \triangleright 2$

(1)

The cell Peclet number is defined as:

$Pe= \rho \left| U_{f} \right| A_{f}/D_{f}$

(1)

in which $\boldsymbol{A_{f}}$ and $\boldsymbol{D_{f}}$ are respectively, the cell-face area and physical diffusion coefficient. When $\boldsymbol{Pe\triangleright 2}$ ,CDS calculations tends to become unstable so that theHDS reverts to the UDS. Physical diffusion is ignored when $\boldsymbol{Pe\triangleright 2}$ .

The HDS scheme is marginally more accurate than the UDS, because the 2nd-order CDS will be used in regions of low Peclet number.

Power-Law Scheme

Patankar, S. V. (1980), Numerical Heat Transfer and Fluid Flow, ISBN 0070487405, McGraw-Hill, New York.

High Resolution Schemes (HRS)

Classification of High Resolution Schemes

HRS can be classified as linear or non-linear, where linear means their coefficients are not direct functions of the convected variable when applied to a linear convection equation. It is important to recognise that linear convection schemes of 2nd-order accuracy or higher may suffer from unboudedness, and are not unconditionally stable.

Non-linear schemes analyse the solution within the stencil and adapt the discretisation to avoid any unwanted behavior, such as unboundedness (see Waterson [1994]). These two types of schemes may be presented in a unified way by use of the Flux-Limiter formulation (Waterson and Deconinck [1995]), which calculates the cell-face value of the convected variable from:

$\phi_{f}= \phi_{c} + 0.5 B \left( r \right) \left( \phi_{c}-\phi_{u} \right)$

(1)

where $\boldsymbol{B \left( r \right)}$ is termed a limiter function and the gradient ration $\boldsymbol{r}$ is defined as:

$r= \left( \phi_{d} - \phi_{c} \right) / \left( \phi_{c} - \phi_{u} \right)$

(1)

The generalisation of this approach to handle non-uniform meshes has been given by Waterson [1994]

From equation (\ref{eq9}) it can be seen that $\boldsymbol{B=1}$ gives the UDS and $\boldsymbol{B=r}$ gives the CDS.

Please note that linear does not mean first order

Linear schemes

Linear schemes are those for which $\boldsymbol{B}$ is linear function of $\boldsymbol{r}$

$\boldsymbol{B(r) = 0}$ is upwind differencing (first-order accurate)

$\boldsymbol{B(r) = r}$ is central differencing (second-order accurate)

Higher order schemes are usually members of the $\boldsymbol{B \left( \kappa \right)}$ class, for which

$B\left( r \right) = 0.5 \left[ \left( 1 + \kappa \right) r + \left( 1 - \kappa \right) \right]$

(1)

Using this equation face variable can be expressed:

in usual variabales

$f_{w}=f_{C}+ \frac{1}{4}\left[\left( 1+\kappa \right)\left(f_{D}-f_{C}\right)+\left(1-\kappa \right) \left( f_{D}-f_{U} \right)\right]$

(1)

in normalised variables

$\hat{f_{w}}=\hat{f_{w}}+\frac{1}{4} \left[\left( 1+\kappa \right)\left( 1-\hat{f_{C}}\right)+ \left( 1-\kappa \right)\hat{f_{C}}\right]$

(1)

The main schemes are

$\boldsymbol{\kappa = 1}$ CDS (central differencing scheme)

$\boldsymbol{\kappa = -1}$ QUICK (quadaratic upwind scheme)

$\boldsymbol{\kappa = 0.5}$ LUS (linear upwind scheme)

$\boldsymbol{\kappa = 0 }$ Fromm

$\boldsymbol{\kappa = 1/3}$ CUS (cubic upwind scheme)

Non-Linear schemes

Non-linear schemes are those for which $\boldsymbol{B}$ is not a linear function of $\boldsymbol{r}$ . They fall into four categories, depending on the linear schemes on which they are based.

$\boldsymbol{(a)}$ QUICK based:

SMART (piecewise linear, bounded)

$B\left( r \right) = \max \left( 0, \min \left( 2r, \ 0.75r + 0.25, \ 4 \right) \right)$

(1)

H-QUICK (smooth)

$B\left( r \right) = 2 \left( r + \left| r \right| \right) / \left( r + 3 \right)$

(1)

UMIST (piecewise linear , bounded)

$B\left( r \right) = \max \left( 0, \ \min \left( 2r, \ 0.75r + 0.25, \ 025 r+ 0.75 , 2 \right)\right)$

(1)

CHARM (smooth, bounded)

$B\left( r \right) = r \left( 3r + 1 \right)/\left( r + 1 \right)^{2} \ for \ r \triangleright 0$

(1)

$B\left( r \right) = 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for \ r \triangleleft 0$

(1)

$(b)$ Fromm based:\\

MUSCL (piecewise linear)

$B\left( r \right) = \max \left( 0, \min \left( 2r, 0.5r + 0.5, 2 \right) \right)$

(1)

van Leer (smooth)

$B\left( r \right) = \left( r + \left| r \right| \right) / \left( r + 1 \right)$

(1)

OSPRE (smooth)

$B\left( r \right) = 1.5r \left(r +1 \right) / \left( r^{2} + r + 1 \right)$

(1)

van Albada (smooth)

$B\left( r \right) = r \left( r + 1 \right) / \left( r^{2} + 1 \right)$

(1)

$(c)$ other:\\

Superbee (piecewise linear)

$B\left( r \right) = \max \left(0, \min \left( 2r , 1 \right), \min \left( r , 2 \right) \right)$

(1)

Minmod (piecewise linear)

$B\left( r \right) = \max \left( 0 , \min \left( r , 1 \right) \right)$

(1)

Waterson, N. P and Deconinck, H (1995), "A unified approach to the desing and application of bounded high-order covection schemes", VKI preprint 1995-21.

Waterson, N. P. (1994), "Development of bounded high-order convection scheme for general industrial applications", VKI Project Report 1994-33.

Numerical Implementation of HRS

The HRS schemes can be introduced into equation (\ref{eq4b}) by using the deffered correction procedure of Rubin and Khosla [1982]. This procedure express the cell-face value $\boldsymbol{\phi_{f}}$ by:

$\phi_{f}=\phi_{f}\left(U \right) + \phi^{'}_{f}$

(1)

where $\boldsymbol{\phi^{'}_{f}}$ is a higher-order correction which represents the difference between the UDS face value $\boldsymbol{\phi_{f}\left(U \right)}$ and the higher-order scheme value $\boldsymbol{\phi_{f}\left(H \right)}$ , i.e.

$\phi^{'}_{f}= \phi_{f}\left(H \right) + \phi_{f}\left(U \right)$

(1)

If equation (\ref{eq10a}) is substituted into equation (\ref{eq4b}), the resulting discretised equation is:

$\begin{matrix} \left\{ D_{h} + C_{h} \phi_{h} \left( U \right) \right\} - \left\{ D_{l} + C_{l} \phi_{l} \left( U \right) \right\} & + & \\ \left\{ D_{n} + C_{n} \phi_{n} \left( U \right) \right\} - \left\{ D_{s} + C_{s} \phi_{s} \left( U \right) \right\} & + & \\ \left\{ D_{e} + C_{e} \phi_{e} \left( U \right) \right\} - \left\{ D_{w} + C_{w} \phi_{w} \left( U \right) \right\} & &= S_{p} + B_{p} \end{matrix}$

(1)

where $\boldsymbol{B_{p}}$ is the deferred-correction source terms, given by:

$B_{p} = C_{l}\phi^{'}_{l} - C_{h}\phi^{'}_{h} + C_{s}\phi^{'}_{s} - C_{n}\phi^{'}_{n} + C_{w}\phi^{'}_{w} - C_{e}\phi^{'}_{e}$

(1)

This treatment leads to a diagonally dominant coefficient matrix since it is formed using the UDS.

The final form of the discretised equation:

$\begin{matrix} a_{P}\phi_{P}= & & a_{N}\phi_{N} &+& a_{S}\phi_{S} &+& a_{E}\phi_{E} \\ & + & a_{W}\phi_{W} &+& a_{H}\phi_{H} &+& a_{L}\phi_{L} \\ & + & a_{T}\phi_{T} &+& S_{p} &+& B_{p} \end{matrix}$

(1)

Subscrit $\boldsymbol{P}$ represents the current computational cell; $\boldsymbol{N, S, E, W, H, L}$ represent the six neighbouring cells and $\boldsymbol{T}$ represents the previous timestep (transistent cases only)

The coefficients contain the appropriate contributions from the transient, convective and diffusive terms in (\ref{eq1})

S.G.Rubin and P.K.Khoshla (1982), "Polynomial interpolation method for viscous flow calculations", J. Comp. Phys., Vol. 27, pp. 153.

Normalised Variables Formulation (NVF)

B.P.Leonard (1988), "Simple high-accuracy resolution program for convective modelling of discontinuities", International J. Numerical Methods Fluids, 8:1291-1318.

Normalised Variable and Space Formulation (NVSF)

Darwish M.S. and Moukalled F., Normalized Variable and Space Formulation Methodology for High-Resolution Schemes, Num. Heat Trans., part B, vol. 26, pp. 79-96, 1994.

Normalised Variables Diagram (NVD)

Total Variation Diminishing (TVD)

Total Variation Diminishing Diagram (Sweby diagram)

Convection Boundedness Criterion (CBC)

Choi S.K., Nam H.Y. and Cho M. (1995), "A comparison of high-order bounded convection schemes", Computational Methods in Applied Mechanics and engineering, Vol. 121, pp. 281-301.

Gaskell P.H. and Lau A.K.C. (1988), "Curvative-compensated convective transport: SMART, a new boundedness-preserving trasport algorithm", International Journal for Numerical Methods in Fluids, Vol. 8, No. 6, pp. 617-641.

Gaskel and Lau have formulated the CBC as follows. A numerical approximation to $\hat{f_{w}}$ is bounded if:

for $0 \leq \hat{f_{C}} \leq 1$ , $f$ is bounded below by the function $\hat{f_{w}} = \hat{f_{W}}$ and above by unity and passes through the points (0,0) and (1,1)

for $\hat{f_{W}} \triangleleft 0$ or $\hat{f_{W}} \triangleright 1$ , $f$ is equal to $\hat{f_{W}}$

The CBC is clearly illustrated in figure below, where the line $\hat{f_{w}} = \hat{f_{W}}$ and the shaded area are the region over which the CBC is valid. The importance of the CBC is to provide a sufficient and necessary condition for guaranteeing the bounded solution if at most three neighbouring nodal values are used to approximate face values. It is well known that the positivity of finite-difference coefficients is also a sufficient condition for boundedness, but this is overly stringent, for the existense of negative coefficients does not neccesarily lead to over- or undershoots.

Schemes

Linear

SOU - Second Order Upwind (also LUDS)

S.P.Vanka ({{{year}}}), "Second-order upwind differencing ina recirculating flow", AIAA J., 25, 1435-1441.

Skew - Upwind

G.D.Raithby , Skew upstream differencing schemes for problems involving fluid flow, Computational Methods Applied Mech. Engineering, 9, 153-164 (1976)

QUICK - Quadratic Upwind Interpolation for Convective Kinematics

B.P.Leonard, A stable and accurate modelling procedure based on quadratic interpolation, Comput. Methods Appl. Mech. Engrg. 19 (1979) 58-98

LUS - Linear Upwind Scheme

H.C.Price, R.S. Varga and J.E.Warren , Application of oscillation matrices to diffusion-convection equations, Journal Math. and Phys., Vol. 45, p.301, (1966)

Fromm - Fromm's Upwind Scheme

CUDS - Cubic Upwind Difference Scheme (also CUS)

Non-Linear QUICK based

SMART - Sharp and Monotonic Algorithm for Realistic Transport

P.H.Gaskell and A.C.K. Lau, Curvature-compensated convective transport: SMART, a new boundedness preserving transport algorithm, International J. Numer. Methods Fluids 8 (1988) 617-641

SMARTER - SMART Efficiently Revised

CHARM - Cubic / Parabolic High-Accuracy Resolution Method

G.Zhou , Numerical simulations of physical discontinuities in single and multi-fluid flows for arbitrary Mach numbers, PhD Thesis, Chalmers University of Technology, Sweden (1995)

UMIST - Upstream Monotonic Interpolation for Scalar Transport

F.S.Lien and M.A.Leschziner , Upstream Monotonic Interpolation for Scalar Transport with application to complex turbulent flows, International Journal for Numerical Methods in Fluids, Vol. 19, p.257, (1994)

Fromm based

MUSCL - Monotonic Upwind Scheme for Conservation Laws

Lien F.S. and Leschziner M.A. , Proc. 5th Int. IAHR Symp. on Refind Flow Modelling and Turbulence Measurements, Paris, Sept. 1993

van Leer limiter

van Albada

OSPRE

Chakravarthy-Osher limiter

Sweby \Phi - limiter

Superbee

MINMOD

Harten A. High resolution schemes using flux limiters for hyperbolic conservation laws. Journal of Computational Physics 1983; 49: 357-393

ISNAS - Interpolation Scheme which is Nonoscillatory for Advected Scalars

Third-order flux-limiter scheme

M. Zijlema , On the construction of a third-order accurate monotone convection scheme with application to turbulent flows in general domains. International Journal for numerical methods in fluids, 22:619-641, 1996.

SOUCUP - Second-Order Upwind Central differnce-first order UPwind

Zhu J. On the higher-order bounded discretization schemes for finite volume compuyations of incompressible flows // Computational Methods in Applied Mechanics and Engineering. 1992. 98. 345-360

HLPA - Hybrid Linear / Parabolic Approximation

Zhu J. Low Diffusive and oscillation-free convection scheme // Communications and Applied Numerical Methods. 1991. 7, N3. 225-232.

Zhu J., Rodi W. A low dispersion and bounded discretization schemes for finite volume computations of incompressible flows // Computational Methods for Applied Mechanics and Engineering. 1991. 92. 87-96

In this scheme, the normalized face value is approximated by a combination of linear and parabolic charachteristics passing through the points, O, Q, and P in the NVD. It satisfies TVD condition and is second-order accurate

Usual variables

$f_{w}= \begin{cases} f_{w} + \left( f_{P} - f_{W} \right) \hat{f_{C}} & 0 \leq \hat{f_{C}} \leq 1 \\ f_{W} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1 \end{cases}$

(2)

Normalized variables - uniform grids

$\hat{f_{w}}= \begin{cases} \hat{f_{C}} \left( 2 - \hat{f_{C}} \right) \hat{f_{C}} & 0 \leq \hat{f_{C}} \leq 1 \\ \hat{f_{C}} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1 \end{cases}$

(2)

Normalized variables - non-uniform grids

$\hat{f_{w}}= \begin{cases} a_{w} + b_{w} \hat{f_{C}} + c_{w} \hat{f_{C}}^{2} & 0 \leq \hat{f_{C}} \leq 1 \\ \hat{f_{C}} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1 \end{cases}$

(2)

where

$a_{w} = 0 , b_{w} = \left(y_{Q}- x^{2}_{Q} \right) / \left(x_{Q}- x^{2}_{Q} \right) , c_{w} = \left(y_{Q}- x_{Q} \right) / \left(x_{Q}- x^{2}_{Q} \right) ,$

(2)

Implementation

Using the switch factors:

for $\boldsymbol{U_w \geq 0}$

$\alpha^{+}_{w} = \begin{cases} 1 & \ if \ | \phi_{P} - 2 \phi_{W} + \phi_{WW}| \triangleleft | \phi_{P} - \phi_{WW} | \\ 0 & otherwise \end{cases}$

(2)

for $\boldsymbol{U_w \triangleleft 0}$

$\alpha^{-}_{w} = \begin{cases} 1 & \ if \ | \phi_{W} - 2 \phi_{P} + \phi_{E}| \triangleleft | \phi_{W} - \phi_{E} | \\ 0 & otherwise \end{cases}$

(2)

and taken all the possible flow directions into account, the un-normalized form of equation can be written as

$\phi_{w} = U^{+}_{w} \phi_{W} + U^{-}_{w} \phi_{P} + \Delta \phi_{w}$

(2)

where

$\Delta \phi_{w} = U^{+}_{w} \alpha^{+}_{w} \left( \phi_{P} - \phi_{W} \right) \frac{\phi_{W} - \phi_{WW}}{\phi_{P} - \phi_{WW}} + U^{-}_{w} \alpha^{-}_{w} \left( \phi_{W} - \phi_{P} \right) \frac{\phi_{P} - \phi_{E}}{\phi_{W} - \phi_{E}}$

(2)

$U^{+}_{w} = 0.5 \left( 1 + \left| U_{w} \right| / U_{w} \right) \ , \ U^{-}_{w} = 1 - U^{+}_{w} \ \ \left( U_{w}\neq 0 \right)$

(2)

CLAM - Curved-Line Advection Method

Van Leer B. , Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. Journal of Computational Physics 1974; 14:361-370

SHARP - Simple High Accuracy Resolution Program

B.P.Leonard, Simple high-accuracy resolution rogram for convective modelling of discontinuities, International J. Numerical Methods Fluids 8 (1988) 1291-1381

LPPA - Linear and Piecewise / Parabolic Approximasion

VONOS - Variable-Order Non-Oscillatory Scheme

Varonos A., Bergeles G., Development and assessment of a Variable-Order Non-oscillatory Scheme for convection term discretization // International Journal for Numerical Methods in Fluids. 1998. 26, N 1. 1-16

CUBISTA - Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection

M.A. Alves, P.J.Oliveira, F.T. Pinho, A convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection // International Lournal For Numerical Methods in Fluids 2003, 41; 47-75

@@ Line 187: / Line 187: @@
 :<math>
 B\left( r \right) = 0.5 \left[ \left( 1 + \kappa \right) r + \left(  1 - \kappa \right) \right]
+</math>
+</td><td width="5%">(1)</td></tr></table>
+Using this equation face variable can be expressed:
+in usual variabales
+<table width="100%"><tr><td>
+:<math>
+f_{w}=f_{C}+ \frac{1}{4}\left[\left( 1+\kappa \right)\left(f_{D}-f_{C}\right)+\left(1-\kappa \right) \left( f_{D}-f_{U} \right)\right]
+</math>
+</td><td width="5%">(1)</td></tr></table>
+in normalised variables
+<table width="100%"><tr><td>
+:<math>
+\hat{f_{w}}=\hat{f_{w}}+\frac{1}{4}
+\left[\left( 1+\kappa \right)\left( 1-\hat{f_{C}}\right)+
+       \left( 1-\kappa \right)\hat{f_{C}}\right]
 </math>
 </td><td width="5%">(1)</td></tr></table>
@@ Line 211: / Line 231: @@
 ==== Non-Linear schemes ====
-Non-linear schemes are those for which $B$ is not a linear function of $r$. They fall into four categories, depending on the linear schemes on which they are based.
+Non-linear schemes are those for which <math>\boldsymbol{B}</math> is not a linear function of <math>\boldsymbol{r}</math>. They fall into four categories, depending on the linear schemes on which they are based.
+* <math>\boldsymbol{(a)}</math> QUICK based:
-$(a)$ QUICK based:\\
 SMART (piecewise linear, bounded)
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) =  \max \left( 0, \min \left( 2r, \ 0.75r + 0.25, \ 4  \right) \right)
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>
 H-QUICK (smooth)
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) =   2 \left( r + \left| r \right| \right) / \left( r + 3 \right)
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>
 UMIST (piecewise linear , bounded)
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) =  \max \left( 0, \ \min \left( 2r, \ 0.75r + 0.25, \ 025 r+ 0.75 , 2 \right)\right)
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>
 CHARM (smooth, bounded)
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) =  r \left( 3r + 1 \right)/\left( r + 1 \right)^{2} \ for \ r \triangleright 0
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) =  0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ for \ r \triangleleft 0
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>
 $(b)$ Fromm based:\\
@@ Line 247: / Line 279: @@
 MUSCL (piecewise linear)
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) =  \max \left( 0, \min \left( 2r, 0.5r + 0.5, 2 \right) \right)
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>
 van Leer (smooth)
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) =  \left( r + \left| r \right|  \right) / \left( r + 1 \right)
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>
 OSPRE (smooth)
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) = 1.5r \left(r +1 \right) / \left( r^{2} + r + 1 \right)
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>
 van Albada (smooth)
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) = r \left( r + 1 \right) / \left( r^{2} + 1 \right)
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>
 $(c)$ other:\\
@@ Line 273: / Line 313: @@
 Superbee (piecewise linear)
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) =  \max \left(0, \min \left( 2r , 1 \right), \min \left( r , 2 \right) \right)
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>
 Minmod (piecewise linear)
-\begin{equation}
+<table width="100%"><tr><td>
+:<math>
 B\left( r \right) = \max \left( 0 , \min \left( r , 1 \right) \right)
-\end{equation}
+</math>
+</td><td width="5%">(1)</td></tr></table>

$\boldsymbol{\kappa = 1}$	CDS (central differencing scheme)
$\boldsymbol{\kappa = -1}$	QUICK (quadaratic upwind scheme)
$\boldsymbol{\kappa = 0.5}$	LUS (linear upwind scheme)
$\boldsymbol{\kappa = 0 }$	Fromm
$\boldsymbol{\kappa = 1/3}$	CUS (cubic upwind scheme)