von Neumann stability analysis
E14978
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.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical method
→
numerical analysis technique → stability analysis method → |
| analyzes |
discrete evolution of error modes
→
spectral properties of difference operators → |
| appliesTo |
finite difference methods
→
linear constant-coefficient PDEs → linear partial differential equations → |
| assumes |
linearization of the numerical scheme
→
superposition of Fourier modes → |
| basedOn |
Fourier mode analysis
→
Fourier series decomposition → assumption of periodic boundary conditions → |
| criterion |
all amplification factors must have modulus less than or equal to one for stability
→
no Fourier mode may grow unbounded in time → |
| field |
computational mathematics
→
numerical analysis → partial differential equations → |
| goal |
determine stable ranges of discretization parameters
→
ensure numerical solution does not grow without bound → |
| historicalContext |
developed in the mid-20th century
→
|
| keyConcept |
Fourier mode
→
amplification factor → stability criterion → |
| limitation |
does not directly address nonlinear stability
→
primarily applicable to linear problems → |
| namedAfter |
John von Neumann
→
|
| relatedTo |
Courant–Friedrichs–Lewy condition
→
Lax equivalence theorem → energy method for stability → |
| requires |
discrete dispersion relation
→
representation of numerical solution as sum of complex exponentials → |
| typicalOutput |
CFL-type stability condition
→
relationship between time step and spatial grid size → time step restriction → |
| usedFor |
analyzing growth of Fourier modes in numerical methods
→
deriving stability conditions for time-marching schemes → stability analysis of finite difference schemes → stability analysis of numerical schemes for PDEs → |
| usedIn |
computational fluid dynamics
→
computational physics → engineering simulations of PDEs → numerical weather prediction → |
| usedWith |
Crank–Nicolson scheme
→
central difference schemes → explicit time-stepping schemes → implicit time-stepping schemes → upwind finite difference schemes → |
Referenced by (3)
| Subject (surface form when different) | Predicate |
|---|---|
|
John von Neumann
→
|
notableConcept |
|
Courant–Friedrichs–Lewy condition
→
|
relatedConcept |
|
Lax equivalence theorem
→
|
relatedTo |