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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.
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.
Here, using a technique such as Crank–Nicolson or the explicit method: the PDE is discretized per the technique chosen, such that the value at each lattice point is specified as a function of the value at later and adjacent points; see Stencil (numerical analysis);
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.
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} .
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In the Crank–Nicolson method, the temperature is equally dependent on t and t + Δt. It is a second- order method in time and this method is generally used in diffusion problems. Stability criteria
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