DELTA Coefficient

A forward centring of the \(\partial h/\partial x\) term in the numerical scheme has a dissipative influence which can be demonstrated using the Taylor-series. The forward centring is automatically introduced in MIKE 1D when the flow becomes supercritical in a particular section of a reach. Additionally, the centring can be specified by the user for subcritical flows through the centring coefficient DELTA (0.5 <d< 1).

Introducing the d value into the terms gives:

\[ \frac{\partial h}{\partial x} = \delta\frac{h_{j+1}^{n+1} - h_{j-1}^{n+1}}{\Delta 2x} + (1-\delta)\frac{h_{j+1}^{n} - h_{j-1}^{n}}{\Delta 2x} \]

Equation: Spatial derivative of h expanded as a Taylor series in the 6-point Abbot scheme.

Using a Taylor series up to the third derivative, these terms become:

\[ \frac{h_{j+1}^{n+1} - h_{j-1}^{n+1}}{\Delta 2x} = \frac{\delta}{\Delta 2x}\left[ h + \frac{1}{1!}\frac{\Delta 2x}{2}\frac{\partial h}{\partial x} + \frac{1}{2!}\left(\frac{\Delta 2x}{2}\right)^{2}\frac{\partial^{2} h}{\partial x^{2}} + \frac{1}{3!}\left(\frac{\Delta 2x}{2}\right)^{3}\frac{\partial^{3} h}{\partial x^{3}} - h + \frac{1}{1!}\frac{\Delta 2x}{2}\frac{\partial h}{\partial x} + \frac{1}{2!}\left(\frac{\Delta 2x}{2}\right)^{2}\frac{\partial^{2} h}{\partial x^{2}} + \frac{1}{3!}\left(\frac{\Delta 2x}{2}\right)^{3}\frac{\partial^{3} h}{\partial x^{3}} \right]_{j}^{n+1} = \delta\left[ \frac{\partial h}{\partial x} + \frac{\left(\Delta 2x\right)^{2}}{24}\frac{\partial^{3} h}{\partial x^{3}} \right] \]

Equation: Spatial derivative of h expanded as a Taylor series in the 6-point Abbot scheme.

Similarly for the last term:

\[ (1-\delta)\frac{h_{j+1}^{n} - h_{j-1}^{n}}{\Delta 2x} = (1-\delta)\left[\frac{\partial h}{\partial x} + \frac{\left(\Delta 2x\right)^{2}}{24}\frac{\partial^{3} h}{\partial x^{3}}\right]_{j}^{n} \]

Equation: The (1-delta)-weighted backward term of the spatial derivative, expanded as a Taylor series.

In this way the formulation of \(\partial h/\partial x\) in the spatial derivative of h expanded as a Taylor series in the 6-point Abbot scheme becomes (again using a Taylor series):

\[ \frac{\partial h}{\partial x} = \delta\left[ \frac{\partial h}{\partial x} + \frac{1}{1!}\frac{\Delta t}{2}\frac{\partial^{2} h}{\partial x \partial t} + \frac{1}{2!}\left(\frac{\Delta t}{2}\right)^{2}\frac{\partial^{3} h}{\partial x \partial t^{2}} \right]_{j}^{n+\frac{1}{2}} + (1-\delta)\left[ \frac{\partial h}{\partial x} - \frac{1}{1!}\frac{\Delta t}{2}\frac{\partial^{2} h}{\partial x \partial t} + \frac{1}{2!}\left(\frac{\Delta t}{2}\right)^{2}\frac{\partial^{3} h}{\partial x \partial t^{2}} \right]_{j}^{n+\frac{1}{2}} = \left.\frac{\partial h}{\partial x}\right|^{n+\frac{1}{2}} + (2\delta - 1)\frac{\Delta t}{2}\left.\frac{\partial^{2} h}{\partial x \partial t}\right|^{n+\frac{1}{2}} + \text{higher order terms} \]

Equation: The DELTA-weighted spatial derivative expanded as a Taylor series and simplified.

which represents a dissipative effect, when d>0.5, as the last term before the higher order terms becomes positive.

Also note that for d>0.5, the solution scheme is not second order, since 2d-1 not zero in equation (A.2.67).