Stability for time-stepping methods is about how numerical errors propagate along a computed trajectory.
Test equation
Definition. The scalar test equation \(\mathrm{d}y/\mathrm{d}t=\lambda y\) is used to study linear stability. A method produces \(y_{n+1}=R(h\lambda)y_n\), where \(R\) is the stability function.
Absolute stability region
Definition. The absolute stability region is \(\{z\in\mathbb C: |R(z)|\le1\}\). For decaying modes with \(\operatorname{Re}\lambda\lt0\), the product \(h\lambda\) should lie in this region.
Stiffness
Definition. A problem is stiff when stability forces explicit methods to take much smaller steps than accuracy alone would require. Stiffness is therefore a relation among the equation, the method, and the desired accuracy.
A-stability
Theorem. A method is A-stable if its stability region contains the entire left half-plane. Backward Euler is A-stable; no explicit Runge-Kutta method is A-stable.
Implicit cost
Remark. Implicit methods improve stability by requiring nonlinear or linear solves at each step. The practical method is therefore a coupled object: time discretization plus equation solver plus error controller.
Absolute stability
Apply a one-step method to \(y'=\lambda y\), \(z=h\lambda\): \[ y_{n+1}=R(z)y_n. \] The absolute-stability region is \[ \mathcal S=\{z\in\mathbb C:|R(z)|\le1\}. \] Explicit Euler has \(R(z)=1+z\), so on the negative real axis it requires \(-2\le h\lambda\le0\). Implicit Euler has \(R(z)=(1-z)^{-1}\), containing the entire left half-plane.
A-stability and L-stability
Definition. A method is A-stable if \(\{z:\operatorname{Re}z\le0\}\subset\mathcal S\). It is L-stable if it is A-stable and \(R(z)\to0\) as \(z\to\infty\) in the left half-plane.
Trapezoidal rule has \[ R(z)=\frac{1+z/2}{1-z/2} \] and is A-stable but not L-stable: \(R(z)\to-1\), so very stiff modes can oscillate numerically instead of being damped. Implicit Euler is L-stable.
Stiffness
A problem is stiff when stability, rather than accuracy of the slow components, forces an explicit method to take tiny steps. In a system \(y'=Ay\) with eigenvalues spanning very different negative scales, the stiffness ratio is often approximated by \[ \frac{\max|\operatorname{Re}\lambda_i|} {\min|\operatorname{Re}\lambda_i|}. \] This is heuristic for nonnormal or nonlinear systems; transient growth and Jacobian variation also matter.
Dahlquist barriers
Theorem (second Dahlquist barrier). No explicit linear multistep method is A-stable, and an A-stable linear multistep method has order at most two.
This impossibility motivates implicit Runge--Kutta methods, whose Gauss families are A-stable of arbitrarily high order, and BDF methods, which trade order for large stiff-stability sectors.
Contractivity and algebraic stability
For nonlinear dissipative systems satisfying \[ \langle f(t,y)-f(t,z),y-z\rangle\le0, \] B-stable methods preserve nonexpansiveness: \(\|y_{n+1}-z_{n+1}\|\le\|y_n-z_n\|\). Algebraically stable implicit Runge--Kutta methods have this property. Scalar absolute stability alone does not guarantee nonlinear contractivity.
Solving implicit stages
Implicit methods require nonlinear solves at each step. Newton tolerances should be tight enough that algebraic error stays below time-discretization error. Reusing Jacobians and factorizations reduces cost; inexact Newton--Krylov methods scale to large semidiscrete PDE systems. A nominally stable integrator can fail if stage equations are solved inaccurately.