Stabilitypreserving

Stabilitypreserving (also written stability-preserving) is a term used in numerical analysis and applied mathematics to describe a property of certain transformations, discretizations, or numerical methods that maintain the stability characteristics of the underlying model. It emphasizes that the discrete or transformed system should not introduce instabilities that were not present in the original continuous system.

In ordinary differential equations and dynamical systems, a stabilitypreserving method is one whose discrete solution inherits stability properties from the continuous system. This can include Lyapunov stability, dissipativity, or contractivity, and is often achieved under imposed step-size restrictions or by preserving a discrete Lyapunov function. In the context of partial differential equations, stabilitypreserving schemes aim to respect discrete analogues of maximum principles, positivity, and energy dissipation, ensuring that numerical solutions behave similarly to true solutions.

Common approaches to stabilitypreserving design include choosing time-integration schemes that are A-stable or energy-stable, and selecting spatial discretizations that maintain monotonicity or positivity of the solution. Implicit methods, such as certain backward Euler or BDF schemes, are frequently cited for their stability properties on stiff problems, while carefully constructed explicit-implicit (IMEX) schemes can be stability-preserving for mixed regimes. The analysis of such methods often relies on discrete energy estimates or invariant-region arguments.

Limitations of stabilitypreserving properties may arise from nonlinearity, complex boundary conditions, or requiring restrictive step sizes. Related concepts include energy stability, positivity preservation, monotonicity, and invariant-region preservation.