Search results
Results from the WOW.Com Content Network
A method is L-stable if it is A-stable and () as , where is the stability function of the method (the stability function of a Runge–Kutta method is a rational function and thus the limit as + is the same as the limit as ).
Lyapunov, in his original 1892 work, proposed two methods for demonstrating stability. [1] The first method developed the solution in a series which was then proved convergent within limits. The second method, which is now referred to as the Lyapunov stability criterion or the Direct Method, makes use of a Lyapunov function V(x) which has an ...
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory.
Lyapunov functions are used extensively in control theory to ensure different forms of system stability. The state of a system at a particular time is often described by a multi-dimensional vector. A Lyapunov function is a nonnegative scalar measure of this multi-dimensional state.
This led to the concept of L-stability: a method is L-stable if it is A-stable and | | as . The trapezoidal method is A-stable but not L-stable. The implicit Euler method is an example of an L-stable method. [8]
Stiff equation — roughly, an ODE for which unstable methods need a very short step size, but stable methods do not L-stability — method is A-stable and stability function vanishes at infinity; Adaptive stepsize — automatically changing the step size when that seems advantageous; Parareal-- a parallel-in-time integration algorithm
Some extensions of Liapunov's second method, IRE Transactions on Circuit Theory, CT-7, pp. 520–527, 1960. (PDF Archived 2019-04-30 at the Wayback Machine) Barbashin, E. A.; Nikolai N. Krasovskii (1952). Об устойчивости движения в целом [On the stability of motion as a whole]. Doklady Akademii Nauk SSSR (in Russian).
One such method is the famous Babylonian method, which is given by x k+1 = (x k + 2/x k)/2. Another method, called "method X", is given by x k+1 = (x k 2 − 2) 2 + x k. [note 1] A few iterations of each scheme are calculated in table form below, with initial guesses x 0 = 1.4 and x 0 = 1.42.