Editing Numerical stability (section)

== Stability in numerical differential equations ==
The above definitions are particularly relevant in situations where truncation errors are not important. In other contexts, for instance when solving [[differential equation]]s, a different definition of numerical stability is used.

In [[Numerical methods for ordinary differential equations|numerical ordinary differential equations]], various concepts of numerical stability exist, for instance [[Stiff equation#A-stability|A-stability]]. They are related to some concept of stability in the [[dynamical system]]s sense, often [[Lyapunov stability]]. It is important to use a stable method when solving a [[stiff equation]].

Yet another definition is used in [[numerical partial differential equations]]. An algorithm for solving a linear evolutionary [[partial differential equation]] is stable if the [[total variation]] of the numerical solution at a fixed time remains bounded as the step size goes to zero. The [[Lax equivalence theorem]] states that an algorithm [[Numerical methods for ordinary differential equations#Convergence|converges]] if it is [[Numerical methods for ordinary differential equations#Consistency and order|consistent]] and [[Numerical methods for ordinary differential equations#Stability and stiffness|stable]] (in this sense). Stability is sometimes achieved by including [[numerical diffusion]]. Numerical diffusion is a mathematical term which ensures that roundoff and other errors in the calculation get spread out and do not add up to cause the calculation to "blow up". [[Von Neumann stability analysis]] is a commonly used procedure for the stability analysis of [[finite difference method|finite difference schemes]] as applied to linear partial differential equations.  These results do not hold for nonlinear PDEs, where a general, consistent definition of stability is complicated by many properties absent in linear equations.