Is Crank-Nicolson implicit or explicit?
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable.
What is an implicit method?
Implicit methods attempt to find a solution to the nonlinear system of equations iteratively by considering the current state of the system as well as its subsequent (or previous) time state.
Which is better explicit, implicit or Crank Nicolson?
Compare the three methods explicit, implicit and Crank-Nicolson for the time stepping. u i + 1, j − u i, j Δ t = κ u i, j − 1 − 2 u i, j + u i, j + 1 ( Δ x) 2 u i + 1, j = u i, j + κ Δ t ( Δ x) 2 ( u i, j − 1 − 2 u i, j + u i, j + 1) u → i + 1 = u → i − κ Δ t A n ⋅ u → i
How is the Crank Nicolson method used in numerical analysis?
In numerical analysis, the Crank–Nicolson method is a finite difference method used for numerically solving the heat equation and similar partial differential equations. It is a second-order method in time. It is implicit in time, can be written as an implicit Runge–Kutta method, and it is numerically stable.
Is the Crank-Nicolson method based on the trapezoidal rule?
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 – the simplest example of a Gauss–Legendre implicit Runge–Kutta method – which also has the property of being a geometric integrator.
What are the subscripts in the Crank-Nicolson method?
where C is the concentration of the contaminant and subscripts N and M correspond to previous and next channel. The Crank–Nicolson method (where i represents position and j time) transforms each component of the PDE into the following: C M ⇒ 1 2 ( C M i j + 1 + C M i j ) .