---
title: "MOC; diffusion, dispersion and stability"
subject: "Computational Techniques in Civil Engineering"
chapter: 6
key: "semester_8/computational_techniques_in_civil_engineering:moc-diffusion-dispersion-stability"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/moc-diffusion-dispersion-stability
site: BCT Engineering
---

# MOC; diffusion, dispersion and stability

Computational Techniques in Civil Engineering · Chapter 6: Method of Characteristics

- Asked in: 2077 Chaitra, 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/moc-diffusion-dispersion-stability
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-6

## Question

What is the method of characteristics? Define diffusion, dispersion and stability.

## Short answer

Method of characteristics converts the PDEs into ODEs along dx/dt=plus or minus a. Diffusion is truncation-error smoothing, especially when interpolation is used for off-grid characteristic feet; dispersion is phase-error wave spreading that distorts fronts. Stability requires Courant number a dt/dx less than or equal to 1; C=1 gives no interpolation error.

## Answer

#### 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the method of characteristics do to the unsteady-flow equations?
   A. Replaces the partial differential equations by ordinary differential equations along dx/dt = ±a
   B. Solves them directly as algebraic equations at every grid point at once
   C. Removes friction so that they can be integrated analytically
   D. Converts them into a single steady-state equation
   Answer: A. Along the C+ and C- lines dx/dt = ±a the two PDEs reduce to ODEs, which are integrated by finite differences. (Key point: Idea of MOC)

2. In the MOC, when does numerical diffusion arise and what is its effect?
   A. When a is zero; it creates spurious oscillations
   B. When the characteristic foot falls between grid points and is interpolated; peaks are averaged and flattened
   C. When C = 1; it increases the amplitude of waves
   D. When friction is neglected; it shifts the wave in time
   Answer: B. Linear interpolation at an off-grid foot smooths sharp pressure peaks, reducing amplitude. (Key point: Numerical diffusion)

3. Which condition ensures stability of the MOC with interpolation, and which value gives the most accurate result?
   A. a dt/dx >= 1; best at large values
   B. a dt/dx <= 1; best at C = 1 (characteristics along the diagonal)
   C. a dt/dx <= 0.5; best at C = 0.5
   D. dt/dx <= 1; best at C = 0
   Answer: B. The Courant condition C = a dt/dx <= 1 keeps the foot within the grid interval; C = 1 needs no interpolation and gives no diffusion. (Key point: Stability and Courant number)

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