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.
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 NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
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