---
title: "Order of accuracy, consistency and stability"
subject: "Applied Computational Fluid Dynamics"
chapter: 3
key: "semester_7/applied_computational_fluid_dynamics:accuracy-order-consistency-convergence"
source: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/accuracy-order-consistency-convergence
site: BCT Engineering
---

# Order of accuracy, consistency and stability

Applied Computational Fluid Dynamics · Chapter 3: Discretization Methods

- Asked in: year not given, 5 marks
- Video: [Defining Consistency, Convergence and Stability of any Numerical Methods|| Lecture 18](https://www.youtube.com/watch?v=rR14_izV8Sk) — Muhammad Asif Farooq (PhD)
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/applied_computational_fluid_dynamics/accuracy-order-consistency-convergence
- Whole chapter: https://www.bctengineer.com/md/semester_7/applied_computational_fluid_dynamics/chapter-3

## Question

Explain truncation error and order of accuracy of a finite difference scheme. Distinguish between first-order and second-order schemes, and explain consistency, stability and convergence.

## Short answer

Truncation error is the dropped Taylor series terms when approximating derivatives; leading term proportional to h^p gives order p. First-order schemes (e.g., upwind) have error proportional to h, second-order (e.g., central) proportional to h^2. Consistency: truncation error tends to zero as h,Δt→0. Stability: errors do not grow unbounded. Convergence: numerical solution approaches exact solution as mesh refines. Lax equivalence theorem: for linear well-posed problems, consistent and stable implies convergent.

## Answer

#### Truncation error and order
The **truncation error** is the part of the Taylor series dropped when a derivative is replaced by a difference formula. If the leading dropped term is proportional to $h^p$, the scheme is of **order $p$**, written $O(h^p)$. For small $h$, the error behaves as $E\approx Ch^p$, so halving $h$ reduces the error by a factor $2^p$.

The order is found from two grids by
$$
p=\frac{\ln(E_1/E_2)}{\ln(h_1/h_2)}
$$

#### First order against second order

| Point | First-order scheme | Second-order scheme |
|---|---|---|
| Example | Forward, backward, upwind | Central difference, Crank-Nicolson |
| Error | $\propto h$ | $\propto h^2$ |
| Halving $h$ | Error halves | Error falls to 1/4 |
| Stencil | Smaller (two points) | Larger (three points) |
| Behaviour | Numerical diffusion, but stable and bounded | Less diffusion; may give oscillations (dispersion) |
| Cost for given accuracy | Needs finer mesh | Coarser mesh enough |

#### Consistency, stability, convergence
- **Consistency:** the truncation error tends to zero as $h,\Delta t\to0$; the difference equation then approaches the original PDE.
- **Stability:** errors (round-off) introduced during computation do not grow without bound. Explicit schemes have conditions such as $r=\alpha\Delta t/\Delta x^2\le0.5$ and $C=u\Delta t/\Delta x\le1$.
- **Convergence:** the numerical solution tends to the exact solution as the mesh is refined.

**Lax equivalence theorem:** for a well-posed linear problem, a consistent scheme converges if and only if it is stable.

Other numerical errors include round-off error (finite computer precision) and iteration (convergence) error. A mesh-independence test uses systematic refinement to estimate discretisation error.

## Check yourself

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

1. Truncation error is:
   A. Error from limited computer precision
   B. The part of the Taylor series dropped when approximating a derivative
   C. Error from an unconverged iteration
   D. Error from wrong boundary conditions
   Answer: B. Round-off is a separate error; truncation error is the neglected Taylor terms. (Key point: Truncation error)

2. For a scheme of order p, halving h reduces the error by a factor of:
   A. p
   B. 2p
   C. 2^p
   D. p^2
   Answer: C. E is about C h^p, so halving h divides the error by 2^p. (Key point: Order of accuracy)

3. A stable scheme is one in which:
   A. Truncation error tends to zero
   B. The solution equals the exact one
   C. Mesh is uniform
   D. Errors introduced during computation do not grow without bound
   Answer: D. Stability concerns growth of round-off and other errors; consistency concerns truncation error. (Key point: Stability)

4. The Lax equivalence theorem says that for a well-posed linear problem a consistent scheme:
   A. Converges if and only if it is stable
   B. Is always stable
   C. Is always second order
   D. Converges without stability
   Answer: A. Consistency plus stability is equivalent to convergence. (Key point: Lax equivalence theorem)

## Related in Applied Computational Fluid Dynamics

- [Second derivative central difference scheme](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Asecond-derivative-central-difference) — Applied Computational Fluid Dynamics, chapter 3
- [Numerical check of difference schemes](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Afinite-difference-derivative-numerical) — Applied Computational Fluid Dynamics, chapter 3
- [Explicit, implicit and Crank-Nicolson schemes](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aexplicit-implicit-crank-nicolson) — Applied Computational Fluid Dynamics, chapter 3
- [Upwind versus central scheme for convection](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aupwind-central-convection-numerical) — Applied Computational Fluid Dynamics, chapter 3
- [Derive forward, backward, central differences](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Ataylor-series-finite-differences) — Applied Computational Fluid Dynamics, chapter 3
- [Explicit scheme for transient conduction](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Aexplicit-ftcs-stability-numerical) — Applied Computational Fluid Dynamics, chapter 3

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