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 equations, a different definition of numerical stability is used.
In numerical ordinary differential equations, various concepts of numerical stability exist, for instance A-stability. They are related to some concept of stability in the dynamical systems 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 converges if it is consistent and 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 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.
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