Stiff equation

In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution. y ′ ( t ) = − 15 y ( t ) , t ≥ 0 , y ( 0 ) = 1. {displaystyle ,y'(t)=-15y(t),quad tgeq 0,y(0)=1.}     (1) y ( t ) = e − 15 t {displaystyle y(t)=e^{-15t},} with y ( t ) → 0 {displaystyle y(t) o 0} as t → ∞ . {displaystyle t o infty .}     (2) y n + 1 = y n + 1 2 h ( f ( t n , y n ) + f ( t n + 1 , y n + 1 ) ) , {displaystyle y_{n+1}=y_{n}+{frac {1}{2}}hleft(f(t_{n},y_{n})+f(t_{n+1},y_{n+1}) ight),}     (3)    (4) y ′ = A y + f ( x ) , {displaystyle mathbf {y} '=mathbf {A} mathbf {y} +mathbf {f} (x),}     (5) y ( x ) = ∑ t = 1 n κ t exp ⁡ ( λ t x ) c t + g ( x ) , {displaystyle mathbf {y} (x)=sum _{t=1}^{n}kappa _{t}exp(lambda _{t}x)mathbf {c} _{t}+mathbf {g} (x),}     (6) R e ( λ t ) < 0 , t = 1 , 2 , … , n , {displaystyle Re(lambda _{t})<0,qquad t=1,2,ldots ,n,}     (7) | R e ( λ ¯ ) | ≥ | R e ( λ t ) | ≥ | R e ( λ _ ) | , t = 1 , 2 , … , n {displaystyle |Re({overline {lambda }})|geq |Re(lambda _{t})|geq |Re({underline {lambda }})|,qquad t=1,2,ldots ,n}     (8) | R e ( λ ¯ ) | | R e ( λ _ ) | . {displaystyle {frac {|Re({overline {lambda }})|}{|Re({underline {lambda }})|}}.}     (9) If a numerical method with a finite region of absolute stability, applied to a system with any initial conditions, is forced to use in a certain interval of integration a steplength which is excessively small in relation to the smoothness of the exact solution in that interval, then the system is said to be stiff in that interval. Significant difficulties can occur when standard numerical techniques are applied to approximate the solution of a differential equation when the exact solution contains terms of the form eλt, where λ is a complex number with negative real part. m x ¨ + c x ˙ + k x = 0 , x ( 0 ) = x 0 , x ˙ ( 0 ) = 0 , {displaystyle m{ddot {x}}+c{dot {x}}+kx=0,qquad x(0)=x_{0},qquad {dot {x}}(0)=0,}     (10) A = ( 0 1 − 1000 − 1001 ) , {displaystyle mathbf {A} =left({egin{array}{rr}0&1\-1000&-1001end{array}} ight),}     (11) f ( t ) = ( 0 0 ) , {displaystyle mathbf {f} (t)=left({egin{array}{c}0\0end{array}} ight),}     (12) x ( 0 ) = ( x 0 0 ) , {displaystyle mathbf {x} (0)=left({egin{array}{c}x_{0}\0end{array}} ight),}     (13) | − 1000 | | − 1 | = 1000 , {displaystyle {frac {|-1000|}{|-1|}}=1000,}     (14) x ( t ) = x 0 ( − 1 999 e − 1000 t + 1000 999 e − t ) ≈ x 0 e − t . {displaystyle x(t)=x_{0}left(-{frac {1}{999}}e^{-1000t}+{frac {1000}{999}}e^{-t} ight)approx x_{0}e^{-t}.}     (15) In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution. When integrating a differential equation numerically, one would expect the requisite step size to be relatively small in a region where the solution curve displays much variation and to be relatively large where the solution curve straightens out to approach a line with slope nearly zero. For some problems this is not the case. Sometimes the step size is forced down to an unacceptably small level in a region where the solution curve is very smooth. The phenomenon being exhibited here is known as stiffness. In some cases we may have two different problems with the same solution, yet problem one is not stiff and problem two is stiff. Clearly the phenomenon cannot be a property of the exact solution, since this is the same for both problems, and must be a property of the differential system itself. It is thus appropriate to speak of stiff systems. Consider the initial value problem