In computational mathematics, a stiff equation is an initial value problem
u ˙ = f ( u ) , u ( 0 ) = u 0 , t ∈ [ 0 , T ] , {\displaystyle {\dot {u}}=f(u)\,,\qquad u(0)=u_{0}\,,\qquad t\in [0,T]\,,}
where f : R d → R d {\displaystyle f:{\mathbb {R} }^{d}\rightarrow {\mathbb {R} }^{d}} , requiring dedicated implicit time stepping methods for its efficient numerical integration. The simplest mathematical characterization of a stiff equation is the necessary condition
T d ( d i v u f ) ( u ) ≪ − 1 . {\displaystyle {\frac {T}{d}}{\big (}{\mathrm {div} }_{u}\,f{\big )}(u)\ll -1\,.}
Since ( d i v u f ) ( u ) = t r a c e ( g r a d u f ) ( u ) = t r a c e f ′ ( u ) {\displaystyle {\big (}{\mathrm {div} }_{u}\,f{\big )}(u)={\mathrm {trace} }({\mathrm {grad} }_{u}\,f)(u)={\mathrm {trace} }\,f'(u)} , where f ′ ( u ) ∈ R d × d {\displaystyle f'(u)\in {\mathbb {R} }^{d\times d}} is the Jacobian matrix of f {\displaystyle f} at the point u {\displaystyle u} , the criterion above is easily evaluated and quantifies stiffness. The criterion is derived, explained and illustrated for nonlinear stiff equations below. For a linear system with constant coefficients u ˙ = A u {\displaystyle {\dot {u}}=Au} , the divergence is constant, making stiffness a global characteristic whose magnitude is related to the time scale T {\displaystyle T} . For a nonlinear system, stiffness usually varies in space and time along the solution trajectory u ( t ) {\displaystyle u(t)} , where the criterion quantifies stiffness locally. In practical computations, stiff equations are invariably solved using adaptive methods.
Background There is a rich literature on stiff differential equations, but intuitive descriptions and heuristics are far more common than attempts at a rigorous definition of the concept. Hairer and Wanner concisely describe the most obvious feature:
Stiff equations are problems for which explicit methods don't work.
This refers to the observation that explicit integration methods are forced to use exceedingly small time steps h {\displaystyle h} to maintain numerical stability, preventing such methods from being competitive. Although each step is inexpensive, the total number of steps N = T / h {\displaystyle N=T/h} becomes prohibitively large, and the integration over [ 0 , T ] {\displaystyle [0,T]} effectively stalls. By contrast, implicit methods for stiff equations require costly "algebraic" equation solving on each step. The extra work per step is offset by superior stability properties, allowing the use of large time steps. Without crippling stability restrictions, the total computational effort is manageable, and the requested accuracy can be achieved without loss of efficiency. For some stiff equations, the efficiency may be several orders of magnitude higher than that of even the best explicit methods.
Analogous efficiency issues in other fields of scientific computing The phenomenon of stiffness is analogous to well-known performance issues in other areas of numerical analysis. For example, in optimization, gradient methods have similar limitations. Using the steepest descent method to minimize a convex functional F ( x ) {\displaystyle F(x)} leads to the iteration
… excerpt ends here. Continue reading the full article.
