MOC; diffusion, dispersion and stability

Method of characteristics

The method of characteristics replaces the two partial differential equations of unsteady flow by ordinary differential equations along the characteristic lines $dx/dt=\pm a$ in the $x$-$t$ plane (the $C^+$ and $C^-$ lines). These are integrated by finite differences from the known time level to the next. It is widely used for water hammer.

Numerical diffusion

Smoothing of the solution (loss of amplitude) caused by the truncation error of the scheme. In the MOC it arises when the characteristic foot falls between grid points and $Q,H$ are found by linear interpolation: sharp pressure peaks are averaged and flattened.

Numerical dispersion

Spreading of waves caused by phase errors: different frequency components travel at different speeds, so a steep wave front becomes distorted and may show spurious oscillations. It is a property of the discretisation, not of the physical pipe.

Stability

A scheme is stable if errors made at one step do not grow in later steps. For the MOC with interpolation, the Courant condition must hold:

$$C=\frac{a\,\Delta t}{\Delta x}\le1$$

so that the characteristic foot lies within the grid interval used for interpolation. $C=1$ (characteristics along the grid diagonal) gives no interpolation error, no numerical diffusion and the most accurate result.