NormConserving

Normconserving is an expression used in numerical analysis and computational physics to describe methods or constructs that preserve a specified norm under discrete approximation or evolution. In numerical analysis, norm-conserving schemes are designed to maintain the norm of a solution, such as the L2 norm for time-dependent problems or the L1 norm for mass conservation, across time steps or spatial discretizations. This property supports stability and physical fidelity, and is achieved in various ways, including unitary time integrators for linear equations, and conservative discretizations for advection, diffusion, or wave propagation. Spectral and finite-difference methods are often formulated to conserve chosen norms or invariants.

In quantum chemistry and condensed matter physics, the term is widely used in the context of norm-conserving pseudopotentials. These are effective core potentials that reproduce, outside a defined core region, the norm (integrated charge) of the all-electron valence wavefunctions. Developed to maintain scattering properties and transferability, norm-conserving pseudopotentials enable efficient calculations by removing core electrons while preserving physical accuracy. Notable families include the Troullier-Martins and Hamann-type pseudopotentials; alternatives such as ultrasoft pseudopotentials and projector augmented-wave methods relax the norm condition for efficiency.

Applications and trade-offs: norm-conserving methods tend to be robust and physically faithful but may impose stricter construction or larger basis sets, whereas norm-relaxing methods can offer computational savings at some cost to strict norm preservation.