---
title: "Numerical dispersion, stability; four-point implicit (600 m)"
subject: "Computational Techniques in Civil Engineering"
chapter: 5
key: "semester_8/computational_techniques_in_civil_engineering:numerical-dispersion-four-point-b4"
source: https://www.bctengineer.com/q/semester_8/computational_techniques_in_civil_engineering/numerical-dispersion-four-point-b4
site: BCT Engineering
---

# Numerical dispersion, stability; four-point implicit (600 m)

Computational Techniques in Civil Engineering · Chapter 5: Finite difference method

- Asked in: 2074 Bhadra, 4+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/numerical-dispersion-four-point-b4
- Whole chapter: https://www.bctengineer.com/md/semester_8/computational_techniques_in_civil_engineering/chapter-5

## Question

Describe numerical dispersion, diffusion and stability of finite difference schemes. The value of flow rate $Q$ at four points in the space-time grid are shown in the figure below. Determine the value of first-order derivatives $\partial Q/\partial t$ and $\partial Q/\partial x$ by using the four-point implicit method. Given: $\Delta t = 1$ hour, $\Delta x = 600$ m and $\theta = 0.55$.

[Figure: space-time grid with time levels n and n+1 and distance points i and i+1; $Q$ at (i, n+1) $= 95$ m$^3$/s, at (i+1, n+1) $= 92$ m$^3$/s, at (i, n) $= 93$ m$^3$/s, at (i+1, n) $= 89$ m$^3$/s]

## Short answer

Numerical diffusion smears fronts due to even-order truncation error; dispersion causes oscillations from odd-order phase error. Stability requires error not to grow (Courant condition for explicit, implicit unconditionally stable for θ≥0.5). With the four-point implicit method, ∂Q/∂t = 6.94×10⁻⁴ m³/s², ∂Q/∂x = −0.00575 m²/s.

## Answer

#### Numerical diffusion
The leading truncation error of an odd-order scheme (e.g. first-order upwind) behaves like a physical diffusion term. For $Q_t+cQ_x=0$ with backward-space, forward-time:
$$
Q_t+cQ_x=\frac{c\Delta x}{2}(1-C)\,Q_{xx}+\dots
$$
The solution is smeared: peaks are lowered and sharp fronts spread out, although the true equation has no diffusion.

#### Numerical dispersion
For second-order (central) schemes the leading error term has an odd derivative, $\propto Q_{xxx}$. Different wavelengths then travel at different speeds, so a sharp front produces spurious oscillations (ripples ahead of or behind the front). It is a phase error, with little change in amplitude.

#### Stability
A scheme is stable if errors (round-off or truncation) do not grow from step to step. For explicit schemes, the von Neumann or Courant-Friedrichs-Lewy condition requires
$$
C=\frac{c\,\Delta t}{\Delta x}\le1
$$
Implicit schemes (and the four-point scheme with $\theta\ge0.5$) are unconditionally stable, but a large $\Delta t$ still reduces accuracy.

#### Four-point implicit method
$$
\frac{\partial Q}{\partial t}\approx\frac{(Q_i^{n+1}-Q_i^n)+(Q_{i+1}^{n+1}-Q_{i+1}^n)}{2\Delta t}
$$
$$
\frac{\partial Q}{\partial x}\approx\theta\,\frac{Q_{i+1}^{n+1}-Q_i^{n+1}}{\Delta x}+(1-\theta)\,\frac{Q_{i+1}^n-Q_i^n}{\Delta x}
$$
Data: $Q_i^{n+1}=95$, $Q_{i+1}^{n+1}=92$, $Q_i^n=93$, $Q_{i+1}^n=89$; $\Delta t=3600$ s, $\Delta x=600$ m, $\theta=0.55$.

**Time derivative**
$$
\frac{\partial Q}{\partial t}=\frac{(95-93)+(92-89)}{2\times3600}=\frac{5}{7200}=6.9444\times10^{-4}\ \text{m}^3/\text{s}^2
$$
**Space derivative**
$$
\frac{\partial Q}{\partial x}=0.55\,\frac{92-95}{600}+0.45\,\frac{89-93}{600}=-0.00275+(-0.00300)=-0.00575\ \text{m}^3/\text{s per m}
$$
**Answer:** $\partial Q/\partial t=6.9444\times10^{-4}$ m$^3$/s$^2$ and $\partial Q/\partial x=-0.00575$ m$^2$/s.

## Check yourself

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

1. Numerical diffusion of a scheme shows up as:
   A. spurious oscillations near a front with no amplitude change
   B. smearing: lowered peaks and spread-out sharp fronts
   C. an unstable growth of round-off error
   D. exact translation of the wave
   Answer: B. The leading truncation error acts like a Qxx term, smoothing the solution. (Key point: Numerical diffusion)

2. Numerical dispersion mainly arises from:
   A. an even-derivative truncation term giving damping
   B. an odd-derivative term (Qxxx) so different wavelengths travel at different speeds
   C. friction in the channel
   D. boundary condition errors
   Answer: B. It is a phase error causing ripples near fronts. (Key point: Numerical dispersion)

3. The CFL condition for explicit schemes requires:
   A. C = c dt/dx <= 1
   B. C >= 1
   C. dt = 0
   D. theta < 0.5
   Answer: A. If C exceeds 1 errors grow for explicit schemes. (Key point: Stability)

4. With theta = 0.55, dx = 600 m, the four-point dQ/dx is:
   A. +0.00575
   B. -0.00575
   C. -0.0030
   D. -0.0115
   Answer: B. 0.55(92-95)/600 + 0.45(89-93)/600 = -0.00275 - 0.00300 = -0.00575. (Key point: Space derivative)

## Close variants set in other exams

- [Numerical dispersion, stability; four-point implicit (500 m)](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Anumerical-dispersion-four-point-j) — Computational Techniques in Civil Engineering, chapter 5, asked 2×

## Related in Computational Techniques in Civil Engineering

- [Four-point implicit derivatives of Q](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afour-point-implicit-numerical-m0) — Computational Techniques in Civil Engineering, chapter 5
- [First and second order finite difference schemes](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afd-first-second-order-schemes) — Computational Techniques in Civil Engineering, chapter 5, asked 2×
- [Finite difference method; explicit and implicit schemes](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Afdm-explicit-implicit) — Computational Techniques in Civil Engineering, chapter 5, asked 2×