##  [Consistency](/consistency-0) 

 Definition

Property of a discrete or approximate formulation that its local formulation reproduces the governing relations of the exact (often continuous) problem in the limit of ideal refinement; commonly expressed as the local truncation error tending to zero as mesh size or step size tends to zero.

 

 

 

 

 

 





## Principle

Principle

A method is consistent if the difference between the discrete operator and the continuous operator applied to sufficiently smooth functions vanishes under ideal refinement; consistency is a necessary ingredient for convergence but not sufficient without stability.

 

 

 

 

 





## Demonstration

Demonstration

A finite difference approximation of the first derivative using a centered three-point stencil: applying the Taylor expansion shows the truncation error is O(h^2), which tends to zero as the mesh spacing h→0, demonstrating consistency of the discretization.

 

 

 

 

## Misapplication

Misapplication

Assuming that demonstrating consistency alone guarantees that numerical solutions converge to the correct solution on a given mesh; overlooking the need to verify stability or to account for boundary approximation errors.

 

 

 

 

 





## Consequence

Consequence

A consistent discretization ensures that, provided the scheme is stable, the numerical solution can converge to the true solution as the refinement parameter goes to its limit; it thereby legitimizes refinement as a route to accuracy.

 

 

 

 

## Reversal

Reversal

Inconsistency: the discrete equations do not approximate the continuous relations in the refinement limit, so even arbitrarily fine discretizations can retain a finite bias and prevent convergence to the exact solution.

 

 

 

 

 





## Boundary

Boundary

Applies to formal discretizations and approximate formulations derived from exact equations; does not quantify the rate of convergence or guarantee stability, and classical consistency notions assume sufficient smoothness of the target solution.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Consistency is sometimes conflated with convergence or stability; unlike convergence (an asymptotic outcome) or stability (response to perturbations), consistency concerns the algebraic match of operators under refinement.

 

 

 

 

 





## Synthesis

Synthesis

Consistency is the algebraic compatibility between discrete and exact formulations in the refinement limit; it is the structural prerequisite that, together with stability, permits convergence and ultimately accurate approximations.