Math 2.0

Numerical analysis Definition

Stiff equation

Builds on Ordinary differential equationEuler methodEigenvalueJacobian matrix

Definition

Stiffness is not a single precise condition; the definition below is the standard working one. Let II be an interval and consider an ordinary differential equation y′=f(t,y)y' = f(t, y) with ff continuously differentiable, on II. The equation is stiff on II if explicit methods such as the Euler method need much smaller steps than accuracy requires. The reason is that larger steps make the numerical solution unstable. Consider a linear system y′=Ay+g(t)y' = A y + g(t) with a constant n×nn \times n matrix AA. Assume its eigenvalues λ1,…,λn\lambda_1, \dots, \lambda_n all have Re⁡λi<0\operatorname{Re} \lambda_i < 0. A common measure is then the stiffness ratio max⁡i∣Re⁡λi∣min⁡i∣Re⁡λi∣,\frac{\max_i \abs{\operatorname{Re} \lambda_i}}{\min_i \abs{\operatorname{Re} \lambda_i}}, and the system is called stiff when this ratio is large.

Remarks

Example: y′=−1000 (y−cos⁡t)y' = -1000\,(y - \cos t). On the test equation y′=λyy' = \lambda y, the Euler method gives yk+1=(1+hλ) yky_{k+1} = (1 + h\lambda)\, y_k, which stays bounded only if ∣1+hλ∣≤1\abs{1 + h\lambda} \le 1. For λ=−1000\lambda = -1000 this forces h≤0.002h \le 0.002, although after a short time every solution is close to the smooth particular solution 106106+1cos⁡t+103106+1sin⁡t\frac{10^6}{10^6 + 1} \cos t + \frac{10^3}{10^6 + 1} \sin t, which is nearly cos⁡t\cos t. The implicit Euler method yk+1=(1−hλ)−1yky_{k+1} = (1 - h\lambda)^{-1} y_k is stable for all h>0h > 0 when λ<0\lambda < 0. Non-example: y′=−yy' = -y is not stiff, since its stability bound h≤2h \le 2 is far larger than the steps needed for accuracy. For nonlinear equations one uses the eigenvalues of the Jacobian matrix ∂f/∂y\partial f / \partial y along the solution.

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Differential equation, concept, no evidence levelDifferential equationconceptEigenvalue, concept, no evidence levelEigenvalueconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionGrönwall's inequality, statement, no evidence levelGrönwall's inequalitystatementPartial derivative, concept, no evidence levelPartial derivativeconceptSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionDiscrete Grönwall inequality, statement, no evidence levelDiscrete Grönwall ine…statementInitial value problem, definition, no evidence levelInitial value problemdefinitionJacobian matrix, concept, no evidence levelJacobian matrixconceptEuler method, concept, no evidence levelEuler methodconceptStiff equation, definition, no evidence levelStiff equationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Differential equation concept
  2. Eigenvalue concept
  3. Euclidean space object
  4. Function concept
  5. Interval concept
  6. Ordinary differential equation definition
  7. Total derivative concept
  8. Function of several variables concept
  9. Lipschitz condition definition
  10. Grönwall's inequality statement
  11. Partial derivative concept
  12. Solution of an ordinary differential equation definition
  13. Discrete Grönwall inequality statement
  14. Initial value problem definition
  15. Jacobian matrix concept
  16. Euler method concept
  17. Stiff equation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Stiff_equation
About this page
TagAXND5JRQAZ
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WritingW0 stub
MSC65L04, 34A99
DefinesStiff equation
UsesOrdinary differential equation, Euler method, Eigenvalue, Jacobian matrix
Changed2 November 2026

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