Crank–Nicolson scheme

E87777

finite difference scheme method for partial differential equations numerical method time-stepping scheme

The Crank–Nicolson scheme is a finite difference method for numerically solving time-dependent partial differential equations, especially parabolic ones like the heat equation, known for its second-order accuracy and unconditional stability.

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All labels observed (2)

Label	Occurrences
Crank–Nicolson method	6
Crank–Nicolson scheme canonical	2

How this entity was disambiguated

This entity first appeared as the object of triple T738106 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Crank–Nicolson scheme
Context triple: [von Neumann stability analysis, usedWith, Crank–Nicolson scheme]

A. Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
B. von Neumann stability analysis
Von Neumann stability analysis is a mathematical technique used in numerical analysis to determine the stability of finite difference schemes for solving partial differential equations by examining the growth of Fourier modes.
C. Euler–Maruyama method
The Euler–Maruyama method is a basic time-stepping scheme for numerically approximating solutions to stochastic differential equations, widely used in simulations of systems with noise such as Langevin dynamics.
D. Gauss–Seidel method
The Gauss–Seidel method is an iterative numerical technique used to solve systems of linear equations, particularly in large, sparse problems arising in scientific and engineering computations.
E. Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Crank–Nicolson scheme
Target entity description: The Crank–Nicolson scheme is a finite difference method for numerically solving time-dependent partial differential equations, especially parabolic ones like the heat equation, known for its second-order accuracy and unconditional stability.

A. Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
B. von Neumann stability analysis
Von Neumann stability analysis is a mathematical technique used in numerical analysis to determine the stability of finite difference schemes for solving partial differential equations by examining the growth of Fourier modes.
C. Euler–Maruyama method
The Euler–Maruyama method is a basic time-stepping scheme for numerically approximating solutions to stochastic differential equations, widely used in simulations of systems with noise such as Langevin dynamics.
D. Gauss–Seidel method
The Gauss–Seidel method is an iterative numerical technique used to solve systems of linear equations, particularly in large, sparse problems arising in scientific and engineering computations.
E. Euler–Maclaurin summation formula
The Euler–Maclaurin summation formula is a fundamental result in analysis that connects sums and integrals, providing powerful asymptotic expansions and error estimates for approximating series by integrals.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	finite difference scheme ⓘ method for partial differential equations ⓘ numerical method ⓘ time-stepping scheme ⓘ
accuracyOrderSpace	2 ⓘ
accuracyOrderTime	2 ⓘ
appliedTo	multi-dimensional diffusion equations ⓘ one-dimensional heat equation ⓘ
assumes	sufficient smoothness of the solution for second-order convergence ⓘ
basedOn	trapezoidal rule for time integration ⓘ
category	A-stable linear multistep-like method ⓘ implicit finite difference method ⓘ
comparedWith	backward Euler scheme ⓘ explicit finite difference scheme ⓘ
developedIn	mid-20th century ⓘ
discretizes	spatial derivatives via finite differences ⓘ time derivative ⓘ
field	computational mathematics ⓘ numerical analysis ⓘ
hasAdvantage	higher accuracy than first-order schemes ⓘ larger stable time steps than explicit schemes ⓘ
hasDisadvantage	may exhibit oscillations for sharp gradients ⓘ requires solving implicit equations ⓘ
hasProperty	A-stable ⓘ implicit ⓘ second-order accuracy in space ⓘ second-order accuracy in time ⓘ time-centered ⓘ unconditionally stable for linear diffusion-type problems ⓘ
implementedIn	many scientific computing libraries ⓘ
namedAfter	John Crank ⓘ Phyllis Nicolson ⓘ
relatedTo	theta-method ⓘ trapezoidal rule ⓘ
requires	solution of linear system at each time step ⓘ
specialCaseOf	theta-method with θ = 1/2 ⓘ
stabilityProperty	unconditionally stable for linear parabolic PDEs under standard assumptions ⓘ
taughtIn	graduate-level numerical analysis courses ⓘ numerical PDE courses ⓘ
timeDiscretizationFormula	u^{n+1} - u^{n} = (Δt/2)[L(u^{n+1}) + L(u^{n})] ⓘ
usedFor	heat equation ⓘ parabolic partial differential equations ⓘ time-dependent partial differential equations ⓘ
usedIn	computational fluid dynamics ⓘ diffusion-reaction models ⓘ heat transfer simulations ⓘ option pricing PDEs ⓘ quantitative finance ⓘ

How these facts were elicited