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All Runge–Kutta methods mentioned up to now are explicit methods. Explicit Runge–Kutta methods are generally unsuitable for the solution of stiff equations because their region of absolute stability is small; in particular, it is bounded. [25] This issue is especially important in the solution of partial differential equations.
Diagonally Implicit Runge–Kutta (DIRK) formulae have been widely used for the numerical solution of stiff initial value problems; [6] the advantage of this approach is that here the solution may be found sequentially as opposed to simultaneously.
Fehlberg, Erwin (1969) Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems. Vol. 315. National aeronautics and space administration. Fehlberg, Erwin (1969). "Klassische Runge-Kutta-Nystrom-Formeln funfter und siebenter Ordnung mit Schrittweiten-Kontrolle". Computing. 4: 93– 106.
Explicit examples from the linear multistep family include the Adams–Bashforth methods, and any Runge–Kutta method with a lower diagonal Butcher tableau is explicit. A loose rule of thumb dictates that stiff differential equations require the use of implicit schemes, whereas non-stiff problems can be solved more efficiently with explicit ...
Numerical methods for ordinary differential equations, such as Runge–Kutta methods, can be applied to the restated problem and thus be used to evaluate the integral. For instance, the standard fourth-order Runge–Kutta method applied to the differential equation yields Simpson's rule from above.
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 to gain efficiency by keeping and using the information from previous steps rather than discarding it.
Gauss–Legendre methods are implicit Runge–Kutta methods. More specifically, they are collocation methods based on the points of Gauss–Legendre quadrature. The Gauss–Legendre method based on s points has order 2s. [1] All Gauss–Legendre methods are A-stable. [2] The Gauss–Legendre method of order two is the implicit midpoint rule.
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations.The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satisfies the ...