Discretization replaces an infinite or continuous object by a finite one. The main question is whether the finite objects return to the original problem as resolution increases.
Discretization
Definition. A discretization is a family of finite problems \(\Phi_h(x_h)=y_h\), indexed by a resolution parameter \(h\downarrow0\), together with restriction and reconstruction maps between the continuous and discrete spaces.
Consistency
Definition. A discretization is consistent if the exact continuous solution, when inserted into the discrete equations, leaves a residual that tends to zero as \(h\downarrow0\). The rate of this residual is the local truncation order.
Stability
Definition. A discretization is stable if perturbations in the discrete data or residual produce uniformly controlled perturbations in the discrete solution. Stability is a bound on the inverse discrete operator, not a formal Taylor expansion.
Lax equivalence principle
Theorem. For a well-posed linear initial-value problem and a consistent finite-difference discretization, stability is equivalent to convergence. In short: consistency tells us we approximated the right equation, stability tells us the approximation error is not amplified, and convergence is the consequence.
Meaning of order
Remark. An order statement such as \(O(h^p)\) describes asymptotic behavior under fixed regularity assumptions. It is not a promise that the visible error decreases at rate \(p\) before the mesh is fine enough or after roundoff dominates.
Continuous and discrete problems
Let \(L:X\to Y\) define \(Lu=f\), and let \(L_h:X_h\to Y_h\) define \(L_hu_h=f_h\). Restriction or interpolation maps \(R_h:X\to X_h\), \(S_h:Y\to Y_h\) compare the levels. The local truncation error is \[ \tau_h(u)=L_hR_hu-S_hLu. \] Consistency means \(\tau_h(u)\to0\) for sufficiently regular exact solutions.
Stability converts consistency into convergence
Theorem. Suppose \(L_h\) is invertible and uniformly stable: \[ \|L_h^{-1}\|_{Y_h\to X_h}\le C. \] If \(L_hu_h=S_hf\) and \(Lu=f\), then \[ \|u_h-R_hu\|_{X_h} \le C\|\tau_h(u)\|_{Y_h}. \] Thus consistency implies convergence.
Proof. Subtract: \[ L_h(u_h-R_hu) =S_hf-L_hR_hu=-\tau_h(u). \] Apply the inverse stability bound.
This elementary identity is the common skeleton behind finite differences, finite elements, quadrature, and time stepping. The norms and regularity class must be stated: a scheme can be stable in one norm and unstable in another.
Lax equivalence for linear evolution schemes
Theorem (Lax--Richtmyer, schematic). For a well-posed linear initial-value problem and a consistent linear finite- difference approximation, stability is equivalent to convergence.
The forward implication is the error equation above. For the converse, pointwise boundedness of discrete evolution operators on a dense class follows from convergence; the uniform boundedness principle upgrades it to an operator stability bound. Linearity and well-posedness are essential. Nonlinear schemes need different compactness or monotonicity arguments.
Order and regularity
A method has consistency order \(p\) if \(\|\tau_h(u)\|\le C(u)h^p\). Stability then yields global order \(p\), but only when \(u\) has the derivatives used in the Taylor expansion. Corners, shocks, layers, and nonsmooth coefficients reduce observed order. A formal Taylor calculation is not a convergence theorem without stability and regularity.
Conservation and monotonicity
For nonlinear conservation laws, weak solutions are nonunique unless an entropy condition is imposed. Conservative, consistent, monotone schemes converge to the entropy solution under suitable compactness. Here monotonicity plays the stabilizing role and supplies discrete maximum principles. This is one reason discretizations should preserve structural properties rather than only match derivatives formally.