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Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt ...
Using the big O notation an th-order accurate numerical method is notated as | | u − u h | | = O ( h n ) {\displaystyle ||u-u_{h}||=O(h^{n})} This definition is strictly dependent on the norm used in the space; the choice of such norm is fundamental to estimate the rate of convergence and, in general, all numerical errors correctly.
All are implicit methods, have order 2s − 2 and they all have c 1 = 0 and c s = 1. Unlike any explicit method, it's possible for these methods to have the order greater than the number of stages. Lobatto lived before the classic fourth-order method was popularized by Runge and Kutta.
The name of this class of methods stems from the fact that the correction of the velocity field is computed through the pressure-field. The discretization of this is typically done with either the finite element method or the finite volume method. With the latter, one might also encounter the dual mesh, i.e. the computation grid obtained from ...
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Using an adaptive stepsize is of particular importance when there is a large variation in the size of the derivative. For example, when modeling the motion of a satellite about the earth as a standard Kepler orbit, a fixed time-stepping method such as the Euler method may be sufficient.
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