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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Crank–Nicolson method | 3 |
| 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Crank–Nicolson scheme Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Crank–Nicolson method
this entity surface form:
Crank–Nicolson method
this entity surface form:
Crank–Nicolson method