The part of the Taylor series dropped when approximating a derivative

Error from limited computer precision

Error from an unconverged iteration

Error from wrong boundary conditions

A stable scheme is one in which:

Errors introduced during computation do not grow without bound

The solution equals the exact one

Practice · 5 marks

Order of accuracy, consistency and stability

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.

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.

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.

✓ Answer checked by Claude Opus 5.5 (high effort)

Second derivative central difference scheme · Applied Computational Fluid Dynamics, chapter 3
Numerical check of difference schemes · Applied Computational Fluid Dynamics, chapter 3
Explicit, implicit and Crank-Nicolson schemes · Applied Computational Fluid Dynamics, chapter 3
Upwind versus central scheme for convection · Applied Computational Fluid Dynamics, chapter 3
Derive forward, backward, central differences · Applied Computational Fluid Dynamics, chapter 3
Explicit scheme for transient conduction · Applied Computational Fluid Dynamics, chapter 3
First and second order finite difference schemes · Computational Techniques in Civil Engineering, chapter 5 · asked 2×
Saint-Venant, second-order explicit scheme · Computational Techniques in Civil Engineering, chapter 5 · asked 2×
Second-order scheme, linear kinematic wave · Computational Techniques in Civil Engineering, chapter 5
Finite difference method; explicit and implicit schemes · Computational Techniques in Civil Engineering, chapter 5 · asked 2×

All Applied Computational Fluid Dynamics chapter 3 questions →

Order of accuracy, consistency and stability

Truncation error and order

First order against second order

Consistency, stability, convergence

Related questions