Stiff equation

In computational mathematics, a stiff equation is an initial value problem

${\displaystyle {\dot {u}}=f(u)\,,\qquad u(0)=u_{0}\,,\qquad t\in [0,T]\,,}$

where ${\displaystyle f:{\mathbb {R} }^{d}\rightarrow {\mathbb {R} }^{d}}$, requiring dedicated implicit time stepping methods for its efficient numerical integration. The simplest mathematical characterization of a stiff equation is the necessary condition

${\displaystyle {\frac {T}{d}}{\big (}{\mathrm {div} }_{u}\,f{\big )}(u)\ll -1\,.}$

Since ${\displaystyle {\big (}{\mathrm {div} }_{u}\,f{\big )}(u)={\mathrm {trace} }({\mathrm {grad} }_{u}\,f)(u)={\mathrm {trace} }\,f'(u)}$, where ${\displaystyle f'(u)\in {\mathbb {R} }^{d\times d}}$ is the Jacobian matrix of ${\displaystyle f}$ at the point ${\displaystyle u}$, 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 ${\displaystyle {\dot {u}}=Au}$, the divergence is constant, making stiffness a global characteristic whose magnitude is related to the time scale ${\displaystyle T}$.

For a nonlinear system, stiffness usually varies in space and time along the solution trajectory ${\displaystyle u(t)}$, where the criterion quantifies stiffness locally. In practical computations, stiff equations are invariably solved using adaptive methods.