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

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

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

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

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

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

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

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

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.