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The Crank–Nicolson stencil for a 1D problem. The Crank–Nicolson method is based on the trapezoidal rule, giving second-order convergence in time.For linear equations, the trapezoidal rule is equivalent to the implicit midpoint method [citation needed] —the simplest example of a Gauss–Legendre implicit Runge–Kutta method—which also has the property of being a geometric integrator.
Crank-Nicolson can be viewed as a form of more general IMEX (Implicit-Explicit) schemes. Forward-Backward Euler method The result of applying both the Forward Euler method and the Forward-Backward Euler method for a = 5 {\displaystyle a=5} and n = 30 {\displaystyle n=30} .
In computational statistics, the preconditioned Crank–Nicolson algorithm (pCN) is a Markov chain Monte Carlo (MCMC) method for obtaining random samples – sequences of random observations – from a target probability distribution for which direct sampling is difficult.
The Crank–Nicolson stencil for a 1D problem. In mathematics, especially the areas of numerical analysis concentrating on the numerical solution of partial differential equations, a stencil is a geometric arrangement of a nodal group that relate to the point of interest by using a numerical approximation routine.
Crank–Nicolson method was developed by Crank and Nicolson. [4] Dantzig introduces the simplex method (voted one of the top 10 algorithms of the 20th century) in 1947. [5] Turing formulated the LU decomposition method. [6]
Nicolson's research on heat conduction related to solutions of the heat equation, and with her colleague John Crank she investigated the numerical stability of several solution techniques. The algorithm now known as the Crank–Nicolson method emerged from this work and was published in 1947. [5]
Finite difference methods for heat equation and related PDEs: FTCS scheme (forward-time central-space) — first-order explicit; Crank–Nicolson method — second-order implicit; Finite difference methods for hyperbolic PDEs like the wave equation: Lax–Friedrichs method — first-order explicit; Lax–Wendroff method — second-order explicit
Stencil figure for the alternating direction implicit method in finite difference equations. The traditional method for solving the heat conduction equation numerically is the Crank–Nicolson method. This method results in a very complicated set of equations in multiple dimensions, which are costly to solve.