Stiff equation

A stiff initial value problem in numerical analysis of ordinary differential equations, an initial value problem

With the explicit one-step methods or multistep methods because of their limited stability region have considerable difficulties. This is the case when the constant in the Lipschitz condition ( cf. Theorem of Picard - Lindelöf )

Assumes large values ​​, but the solution is fairly smooth. In this case, numerical methods could approximate this solution exactly with relatively large step sizes, explicit methods are compelled because of the limited stability region to use small step sizes. Typically occur stiff initial value problems in the numerical approximation of parabolic partial differential equations after the discretization in the space domain. One example is the Crank -Nicolson method in which the second-order Adams - Moulton method is used in place a finite difference method and in time direction.

Example

The problem is with the explicit and implicit Euler method and increment by the linear initial value problem

With explained. The exact solution is for large and the solution is almost constant, so very smooth.

  • The explicit Euler method calculated with the approximations
  • In contrast, the implicit Euler method calculated on the basis of the approximations
  • The two graphs show the exact solution in blue, an approximate solution with a small increment in green and the approximate solutions with in red.

The explicit Euler method, the red approximations grow on and on, during these rough approximations remain with the implicit Euler method in the vicinity of the exact solution.

Advanced stability concepts

For a more detailed classification of numerical methods for stiff initial value problems different stability concepts have been introduced in the literature, which are based usually on different test equations. These include the

With numerical methods, it is advantageous if the numerical approximations as the exact solutions behave in test equations substantially. Accordingly, the term calls

  • A- stability, that approximate solutions go in the first test against zero for equations,
  • That remove B- stability, two approximation solutions of the third test equation with one another are not.

For implicit Runge- Kutta methods, there is the term Algebraic stability, a sufficient criterion for B- stability.

Numerical methods for stiff initial value problems

For stiff initial value problems implicit methods are more efficient than explicit ( that you can almost " stiff " View as a definition of the term ). Special classes are

  • Implicit Runge- Kutta methods
  • Rosenbrock - Wanner method
  • BDF methods

Since implicit methods the resolution of non-linear systems of equations requires a lot of effort, even linearly implicit one-step methods such as those mentioned Rosenbrock - Wanner method ( ROW methods) have been developed.

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