Fact-checked by Grok 2 weeks ago

Von Neumann stability analysis

Von Neumann stability analysis is a procedure in numerical analysis for assessing the stability of finite difference approximations to linear partial differential equations with constant coefficients. Developed by mathematician John von Neumann in the mid-1940s, it involves substituting a Fourier mode solution—typically of the form u_j^n = G^n e^{i \kappa j \Delta x}—into the discretized scheme to derive the amplification factor G(\kappa), where stability requires |G(\kappa)| \leq 1 for all wavenumbers \kappa in the interval (-\pi, \pi].^[1]^[2] This method, also known as Fourier stability analysis, originated from von Neumann's unpublished work on numerical solutions to hydrodynamic problems during the Manhattan Project, and was first publicly acknowledged in the 1947 paper by Crank and Nicolson on solving heat conduction equations. It builds on earlier qualitative insights into stability from the 1928 Courant-Friedrichs-Lewy condition, providing a quantitative tool to predict whether small errors in initial data or rounding will grow uncontrollably over time steps.^[3] Key aspects include its applicability to linear schemes on uniform grids, where the analysis decomposes errors into independent Fourier modes and checks amplification across all resolvable wavelengths. For hyperbolic equations like the advection equation, it often yields the Courant-Friedrichs-Lewy (CFL) condition, such as \sigma = c \Delta t / \Delta x \leq 1 for upwind schemes, ensuring information propagates without distortion. In parabolic problems, such as the heat equation, it determines diffusion number restrictions like D = \alpha \Delta t / (\Delta x)^2 \leq 1/2 for explicit methods.^[1]^[2] The technique assumes periodic boundary conditions and ignores boundaries, making it a local sufficient condition for stability; extensions to nonlinear or variable-coefficient cases involve linearization.^[4] Widely used in computational fluid dynamics, weather modeling, and other fields solving time-dependent PDEs, von Neumann analysis complements Lax equivalence theorem, which states that consistency plus stability implies convergence for linear problems. Limitations arise with nonlinearities or stiff systems, where more advanced tools like energy methods or matrix norms may be needed, but it remains a foundational, efficient first step in scheme validation.^[3]^[5]

Introduction

Definition and Purpose

Von Neumann stability analysis is a mathematical procedure that employs Fourier analysis to evaluate the stability of finite difference schemes for solving partial differential equations (PDEs). It decomposes the numerical solution or error into Fourier modes and examines how these modes evolve over successive time steps, determining whether errors grow, decay, or remain bounded. This method is particularly suited to linear PDEs with constant coefficients on uniform grids, where the analysis reduces the problem to scalar ordinary differential equations for each Fourier component.^[3] The primary purpose of Von Neumann stability analysis is to predict whether a numerical scheme will yield bounded solutions for well-posed PDEs, thereby ensuring that small perturbations or rounding errors do not amplify uncontrollably during computation. By deriving an amplification factor for each wave number, the analysis identifies conditions under which the magnitude of these factors remains at most 1 in the limit of small time steps, providing a necessary criterion for stability in infinite or periodic domains that neglect boundary effects. This approach is essential for designing reliable numerical methods in fields like fluid dynamics and heat transfer, where unstable schemes can lead to spurious oscillations or divergence.^[1]^[3] While complementary to the Lax equivalence theorem—which states that for linear PDEs, consistency and stability are equivalent to convergence—Von Neumann analysis specifically offers a practical tool for verifying the stability component without requiring full convergence proofs. It focuses on local error behavior in Fourier space, serving as a necessary but not always sufficient condition for overall scheme reliability.^[3]

Historical Background

The development of Von Neumann stability analysis emerged in the 1940s and 1950s as part of John von Neumann's pioneering efforts in computational mathematics, particularly during his involvement with the ENIAC computer and computations at Los Alamos National Laboratory for simulating fluid dynamics and shock waves in the context of nuclear weapons research.^[3] Von Neumann applied Fourier analysis to assess the stability of finite difference schemes for solving partial differential equations (PDEs), recognizing the need for rigorous criteria to ensure reliable numerical solutions on early electronic computers.^[6] This work built directly on the practical challenges of implementing difference methods for hyperbolic systems, where errors could amplify uncontrollably without proper constraints.^[3] Von Neumann's ideas were first publicly acknowledged in the 1947 paper by Crank and Nicolson, who applied the method to solving the heat conduction equation while crediting von Neumann's unpublished contributions from the mid-1940s. A key document applying von Neumann's approach is the 1950 paper "Numerical Integration of the Barotropic Vorticity Equation" by Charney, Fjörtoft, and von Neumann, published in Tellus. In this paper, the authors detailed methods for time-stepping evolution equations using finite differences, introducing amplification factors derived from Fourier modes to evaluate stability limits.^[7]^[8] The analysis emphasized the role of the Courant number in preventing instability, providing a foundational tool for practitioners in computational physics.^[3] The roots of von Neumann's method trace back to earlier techniques using Fourier series for separation of variables in PDEs, with significant influence from the 1928 paper by Richard Courant, Kurt Otto Friedrichs, and Hans Lewy, which first highlighted stability issues in finite difference approximations for hyperbolic and elliptic equations.^[9] Courant et al. demonstrated that certain schemes fail to converge unless the time step satisfies a condition relating grid spacing and wave speeds, laying the groundwork for von Neumann's more systematic Fourier-based extension. This earlier work, originally published in German as "Über die partiellen Differenzengleichungen der mathematischen Physik," underscored the necessity of stability for well-posedness in numerical PDE solvers.^[9] By the 1950s and 1960s, von Neumann's stability analysis became a standard tool in the burgeoning field of numerical methods, particularly as finite difference techniques proliferated for solving hyperbolic PDEs in aerodynamics, weather prediction, and plasma physics.^[6] Its adoption was accelerated by the formalization in Peter Lax and Robert Richtmyer's 1956 survey, which linked stability to convergence via the Lax equivalence theorem, solidifying its role in ensuring the reliability of simulations on increasingly powerful computers.^[3] This period marked the transition from ad hoc computations to theoretically grounded practices in computational science.^[6]

Numerical Stability Concepts

Overview of Stability in Numerical Methods

In numerical methods for solving differential equations, stability refers to the property that small perturbations in the input data, such as rounding errors or inaccuracies in initial conditions, do not cause the computed solution to deviate significantly from the true solution or grow unboundedly over the course of the computation.^[10] A numerical scheme is considered stable if these perturbations remain controlled and bounded, ensuring that the solution behaves qualitatively similarly to the exact solution of the underlying problem.^[11] Several types of stability are distinguished in the context of ordinary differential equations (ODEs). Zero-stability, introduced by Dahlquist, applies to multistep methods and requires that, as the time step size approaches zero, the numerical solutions remain bounded for any fixed interval, preventing the amplification of errors due to parasitic roots in the method's characteristic polynomial.^[11] For stiff ODEs—systems where solutions include components that decay rapidly due to eigenvalues with large negative real parts—absolute stability is crucial; a method is A-stable if its numerical solutions tend to zero when applied to test equations like \frac{dx}{dt} = qx with q having negative real part, for any fixed positive step size.^[12] Stability can further be classified as unconditional, if it holds for all step sizes, or conditional, if it requires the step size to be sufficiently small relative to problem parameters.^[12] Instability in numerical schemes can lead to severe consequences, including the exponential growth of errors that dominate the solution, resulting in non-physical oscillations, grid-scale artifacts, or complete blow-up of the computation, rendering the results useless even for moderately refined grids.^[10] For instance, parasitic modes in unstable multistep methods can cause the numerical solution to diverge dramatically from the exact one, as observed in high-order explicit schemes where errors amplify uncontrollably.^[11] The importance of stability is underscored by its role in convergence theory: the Lax equivalence theorem establishes that, for linear problems with well-posed continuous equations, a consistent finite difference scheme converges to the true solution if and only if it is stable.^[13] This theorem highlights that stability, alongside consistency (the property that the scheme approximates the differential equation as the discretization parameter vanishes), is necessary and sufficient for reliable numerical approximations. Von Neumann stability analysis serves as a key tool for verifying this property in linear partial differential equation schemes through Fourier mode examination.^[13]

Stability for Partial Differential Equations

Partial differential equations (PDEs) are classified into three main types based on their mathematical structure and physical interpretation: hyperbolic, parabolic, and elliptic. Hyperbolic PDEs, such as the advection equation, model wave-like phenomena where information propagates at finite speeds along characteristics. Parabolic PDEs, exemplified by the heat equation, describe diffusion processes where solutions smooth out over time. Elliptic PDEs, like the Laplace equation, represent steady-state problems without time evolution, often arising in boundary value contexts. Stability requirements in numerical discretizations differ significantly across these classes; hyperbolic and parabolic PDEs demand careful control of time steps to prevent error growth in time-marching schemes, while elliptic PDEs focus on iterative convergence without explicit time dependence.^[14] Discretizing PDEs introduces errors that can propagate uncontrollably, particularly through high-frequency spatial modes that mimic unphysical oscillations. In finite difference methods, spatial approximations may amplify these modes, leading to phenomena like odd-even decoupling, where solutions on even and odd grid points evolve independently, causing checkerboard patterns or stagnation. Temporal discretization exacerbates this if the time step allows errors to grow exponentially, highlighting the need for stability analysis tailored to PDE structures.^[15] For hyperbolic PDEs, a key stability criterion is the Courant-Friedrichs-Lewy (CFL) condition, which conceptually requires the time step \Delta t to satisfy \Delta t \leq \frac{\Delta x}{c}, where \Delta x is the spatial grid size and c is the characteristic wave speed, ensuring that information does not propagate faster than the numerical scheme can resolve. This prevents instability by aligning the numerical domain of dependence with the physical one. Von Neumann stability analysis is particularly valuable for linear constant-coefficient PDEs, as it decomposes errors into Fourier modes to quantify dissipation (amplitude decay) and dispersion (phase speed errors), guiding the selection of stable schemes.^[9]^[1] The Lax equivalence theorem underscores this by stating that, for well-posed linear PDEs, a consistent scheme converges if and only if it is stable.^[16]

The Von Neumann Analysis Method

Fourier Mode Decomposition

In Von Neumann stability analysis, the initial step involves decomposing the numerical solution into Fourier modes to assess the behavior of errors or perturbations in finite difference schemes for partial differential equations (PDEs). This decomposition assumes that the solution on a discrete grid can be represented as a superposition of plane waves, typically expressed as u_j^n = \sum_k \xi_k^n e^{i k j \Delta x}, where u_j^n denotes the solution at spatial index j and time level n, k is the wavenumber, \Delta x is the spatial grid spacing, i = \sqrt{-1}, and \xi_k^n is the amplification factor for mode k at time n.^[17]^[1] This form leverages the exponential nature of Fourier modes, which are eigenfunctions of translation-invariant operators, simplifying the analysis of linear schemes.^[18] For linear numerical schemes with constant coefficients, the separation of variables principle applies, allowing each Fourier mode to evolve independently without interaction between modes. This independence arises because the linearity ensures that the discrete operator acts diagonally in the Fourier basis, treating the problem as if on an infinite domain to ignore boundary effects.^[17]^[1] Consequently, the analysis focuses on a single representative mode u_j^n = \xi^n e^{i k j \Delta x}, where the time dependence is isolated in \xi^n, facilitating the study of how perturbations propagate over time steps.^[18] This approximation is valid under periodic boundary conditions or for sufficiently large domains where boundary influences are negligible.^[1] The continuous PDE, such as \partial u / \partial t = f(u), is first discretized using a finite difference scheme, yielding an update rule like u_j^{n+1} = u_j^n + \Delta t \, L(u_j^n), where \Delta t is the time step and L represents the discrete spatial operator (e.g., a difference approximation to derivatives).^[17] Substituting the Fourier mode into this scheme isolates the spatial effects through the phase factor e^{i k j \Delta x}, which simplifies L to a multiplication by its Fourier symbol.^[1] The wavenumber k is restricted to the range $0 \leq k \leq \pi / \Delta x (or equivalently -\pi / \Delta x \leq k \leq \pi / \Delta x) to encompass all resolvable modes on the grid, corresponding to the Nyquist limit where the shortest wavelength is $2 \Delta x.^[18]^[17] Modes beyond this range alias and are not uniquely representable, ensuring the analysis captures the scheme's behavior for physically meaningful scales.^[1]

Derivation of the Amplification Factor

In Von Neumann stability analysis, the amplification factor \xi is obtained by substituting a Fourier mode solution into the finite difference scheme. Assuming a discrete solution of the form u_j^n = \xi^n e^{i k j \Delta x}, where k is the wavenumber and \Delta x is the spatial grid spacing, this mode represents a single harmonic component propagating without changing shape except for possible amplification or phase shift at each time step.

\] The scheme advances from time level $n$ to $n+1$ via $ \mathbf{u}^{n+1} = A \mathbf{u}^n $, where $A$ is the amplification matrix encoding the discrete operator. Substituting the Fourier mode yields $\xi^{n+1} e^{i k j \Delta x} = A (\xi^n e^{i k j \Delta x})$, implying that $\xi$ is an eigenvalue of $A$ corresponding to the eigenvector $e^{i k j \Delta x}$. This substitution isolates the growth behavior for each mode, as the exponential factors cancel out.\[

For explicit finite difference schemes approximating a partial differential equation of the form u_t = L u, the update is typically u_j^{n+1} = u_j^n + \Delta t \sum_{l} b_l u_{j+l}^n, where the b_l are coefficients of the discrete spatial operator L_h. Substituting the Fourier mode gives

\xi e^{i k j \Delta x} = e^{i k j \Delta x} + \Delta t \sum_{l} b_l e^{i k (j+l) \Delta x} = e^{i k j \Delta x} \left( 1 + \Delta t \sum_{l} b_l e^{i k l \Delta x} \right).

Dividing through by e^{i k j \Delta x} yields the amplification factor

\xi = 1 + \Delta t \, g(\theta),

where \theta = k \Delta x is the dimensionless phase angle and g(\theta) = \sum_{l} b_l e^{i l \theta} is the symbol of the spatial operator L_h, obtained via its Fourier transform.$$] This algebraic form reveals how \xi depends on the time step \Delta t, grid spacing \Delta x, and wavenumber k. The symbol g(\theta) captures the action of L_h in Fourier space. For instance, the central difference approximation to \partial u / \partial x is (u_{j+1} - u_{j-1}) / (2 \Delta x), with symbol g(\theta) = i \sin \theta / \Delta x; alternatively, it can be expressed as i (2 / \Delta x) \sin(\theta / 2) \cdot \cos(\theta / 2), highlighting the exact trigonometric representation.[

For a scheme approximating $u_t + a u_x = 0$ with this spatial discretization, $\xi = 1 - i a \Delta t \sin \theta / \Delta x$. In general, $\xi$ is [complex](/page/Complex), where its [magnitude](/page/Magnitude) $|\xi|$ governs the [exponential growth](/page/Exponential_growth) or [decay](/page/Decay) of the mode [amplitude](/page/Amplitude) over time steps, and its [argument](/page/Argument) $\arg(\xi)$ determines the [phase](/page/Phase) shift, relating to numerical [dispersion](/page/Dispersion) errors in wave propagation.

]

Von Neumann Stability Condition

The Von Neumann stability condition provides a criterion for assessing the stability of finite difference schemes for linear partial differential equations by examining the amplification factor \xi, derived from the scheme's Fourier mode analysis. Specifically, a scheme is deemed von Neumann stable if |\xi(\theta)| \leq 1 + O(\Delta t) for all phase angles \theta = k \Delta x corresponding to wavenumbers k \in [0, \pi / \Delta x], where \Delta t is the time step and \Delta x is the spatial grid spacing.^[3] This bound ensures that errors in the numerical solution do not grow uncontrollably over multiple time steps, as the amplification factor governs the evolution of each Fourier mode from one time level to the next. In the strict sense, many analyses require |\xi| \leq 1 exactly, particularly for single-step schemes, to preclude any amplification beyond neutral propagation.^[1] This condition serves as a necessary requirement to prevent exponential growth of computational errors, which could otherwise dominate the solution as the number of time steps increases. For linear problems with constant coefficients and periodic or infinite domains (i.e., without boundary effects), it is also sufficient for stability in the sense of Lax and Richtmyer, meaning the numerical solution remains bounded independently of the mesh size as \Delta t, \Delta x \to 0. The inclusion of the O(\Delta t) term accommodates slight growth permissible for consistency with the underlying PDE, but violations—such as |\xi| > 1 for any mode—lead to instability where high-frequency errors amplify rapidly.^[3] A key aspect of the condition is the resolution requirement for high-wavenumber modes near the Nyquist frequency (k \approx \pi / \Delta x, or \theta \approx \pi), which represent short-wavelength oscillations resolvable only by the grid. These modes must satisfy |\xi| \leq [1](/page/1) (often |\xi| < [1](/page/1)) to avoid amplification that could manifest as spurious oscillations or grid-scale instabilities in the solution. Failure here typically signals an unstable scheme, as unresolved short waves can dominate error dynamics despite accurate resolution of longer physical modes.^[1] Neutral stability occurs when |\xi| = 1 for low-wavenumber modes (k small, \theta \approx 0), implying no artificial dissipation or dispersion for physically relevant long waves, which preserves the scheme's fidelity to the continuous problem. However, practical stable schemes often introduce controlled dissipation for high-k modes, where |\xi| < 1, to damp numerical noise without overly affecting low-frequency components. This selective behavior ensures overall stability while maintaining accuracy for resolved scales.^[1]

Applications and Illustrations

Example: Linear Advection Equation

The linear advection equation models the transport of a quantity u(x, t) with constant velocity c: [ \frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0.

This hyperbolic partial differential equation describes wave propagation without diffusion or dispersion in the continuous case.[](https://twister.caps.ou.edu/cfd2000/Chapter3a.pdf) A straightforward finite difference discretization is the forward-time centered-space (FTCS) scheme on a uniform grid with spacing $\Delta x$ and time step $\Delta t$:

u_j^{n+1} = u_j^n - \frac{c \Delta t}{2 \Delta x} (u_{j+1}^n - u_{j-1}^n),

where $u_j^n \approx u(j \Delta x, n \Delta t)$. This scheme uses forward differencing for the time derivative and centered differencing for the spatial derivative.[](https://web.stanford.edu/group/frg/course_work/AA214/CA-AA214-Ch6.pdf) Applying Von Neumann stability analysis involves substituting a Fourier mode solution $u_j^n = \xi^n e^{i k j \Delta x}$ into the scheme, assuming periodic boundary conditions and linearity. This yields the amplification factor

\xi = 1 - i \sigma \sin(k \Delta x),

where $\sigma = c \Delta t / \Delta x$ is the Courant number. The magnitude squared is then

|\xi|^2 = 1 + \sigma^2 \sin^2(k \Delta x).

For stability, $|\xi| \leq 1$ must hold for all wavenumbers $k$. However, since $\sin^2(k \Delta x) \geq 0$ and equals zero only for specific modes ($k \Delta x = 0$ or $\pi$), $|\xi|^2 > 1$ for other modes whenever $\sigma > 0$. Thus, the FTCS scheme is unconditionally unstable for any positive $\Delta t$, as high-frequency modes amplify exponentially over time.[](https://www.cs.princeton.edu/courses/archive/fall10/cos323/notes/cos323_f10_lecture19_pde2.pdf)[](https://twister.caps.ou.edu/cfd2000/Chapter3a.pdf) The [instability](/page/Instability) in FTCS stems from dispersive phase errors inherent in the centered spatial differencing, which fails to adequately control the growth of oscillatory modes without dissipative terms.[](https://web.stanford.edu/group/frg/course_work/AA214/CA-AA214-Ch6.pdf) To achieve [stability](/page/Stability), consider the first-order [upwind scheme](/page/Upwind_scheme) (for $c > 0$), which biases the spatial differencing against the flow direction:

u_j^{n+1} = u_j^n - \sigma (u_j^n - u_{j-1}^n).

Substituting the [Fourier](/page/Fourier) [ansatz](/page/Ansatz) gives

\xi = 1 - \sigma (1 - e^{-i k \Delta x}).

The magnitude squared simplifies to

|\xi|^2 = [1 - \sigma (1 - \cos(k \Delta x))]^2 + [\sigma \sin(k \Delta x)]^2 = 1 - 2\sigma (1 - \cos(k \Delta x)) (1 - \sigma).

Since $1 - \cos(k \Delta x) \geq 0$, $|\xi|^2 \leq 1$ for all $k$ if and only if $0 \leq \sigma \leq 1$. This enforces the Von Neumann stability condition via the Courant-Friedrichs-Lewy (CFL) criterion, ensuring no mode amplification. The upwind scheme introduces numerical dissipation that damps short-wavelength errors, promoting stability at the cost of reduced accuracy for smooth solutions.[](https://climate.ucdavis.edu/AOSS605-NumericalMethodsLectures.pdf)[](https://web.stanford.edu/group/frg/course_work/AA214/CA-AA214-Ch6.pdf) ### Example: Heat Equation The one-dimensional [heat equation](/page/Heat_equation), a [parabolic partial differential equation](/page/Parabolic_partial_differential_equation) modeling [diffusion](/page/Diffusion) processes, is given by

\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2},

where $\alpha > 0$ is the [thermal diffusivity](/page/Thermal_diffusivity) coefficient.[](https://web.math.ucsb.edu/~grigoryan/124B/lecs/lec17.pdf) A common explicit [finite difference](/page/Finite_difference) scheme, known as the forward-time centered-space (FTCS) method, discretizes this equation on a uniform grid with spatial step $\Delta x$ and time step $\Delta t$ as

u_j^{n+1} = u_j^n + r (u_{j+1}^n - 2 u_j^n + u_{j-1}^n),

where $ r = \alpha \Delta t / \Delta x^2 $ and $ u_j^n \approx u(j \Delta x, n \Delta t) $.[](https://web.math.ucsb.edu/~grigoryan/124B/lecs/lec17.pdf) This scheme is first-order accurate in time and second-order in space.[](https://math.mit.edu/classes/18.300/Notes/Notes_vNSA.pdf) To apply Von Neumann stability analysis, assume a solution of the form $ u_j^n = \xi^n e^{i k j \Delta x} $, where $ k $ is the wavenumber representing a Fourier mode and $ i = \sqrt{-1} $.[](https://web.math.ucsb.edu/~grigoryan/124B/lecs/lec17.pdf) Substituting into the scheme yields the amplification factor

\xi = 1 - 4 r \sin^2 \left( \frac{k \Delta x}{2} \right).

[](https://web.math.ucsb.edu/~grigoryan/124B/lecs/lec17.pdf) For stability, the condition $ |\xi| \leq 1 $ must hold for all wavenumbers $ k $, ensuring that errors do not grow unbounded over time steps.[](https://math.mit.edu/classes/18.300/Notes/Notes_vNSA.pdf) Since $ \sin^2(\theta) $ ranges from 0 to 1, the minimum $ \xi $ occurs at $ \sin^2 = 1 $, giving $ \xi = 1 - 4r $; requiring $ |1 - 4r| \leq 1 $ and $ \xi \geq -1 $ (which holds for $ r \geq 0 $) leads to the conditional stability criterion $ r \leq 1/2 $, or $ \Delta t \leq (\Delta x^2)/(2 \alpha) $.[](https://web.math.ucsb.edu/~grigoryan/124B/lecs/lec17.pdf) In contrast, the implicit backward-time centered-space (BTCS) scheme for the [heat equation](/page/Heat_equation) is

u_j^{n+1} - r (u_{j+1}^{n+1} - 2 u_j^{n+1} + u_{j-1}^{n+1}) = u_j^n.

[23] Applying the same [Fourier](/page/Fourier) [ansatz](/page/Ansatz) produces the amplification factor

\xi = \frac{1}{1 + 4 r \sin^2 \left( \frac{k \Delta x}{2} \right)},

References

[1]
[PDF] Notes: von Neumann Stability Analysis - MIT Mathematics
Apr 26, 2022 · A von Neumann stability analysis can be carried out for constant coefficients linear finite differences schemes only. It is based on the fact ...
[2]
7.4 Stability analysis with von Neumann's method
The von Neumann analysis is commonly used to determine stability criteria as it is generally easy to apply in a straightforward manner.
[3]
[PDF] Chapter 4. Accuracy, Stability, and Convergence - People
The first is by Fourier or von Neumann analysis, applicable in cases where one is dealing with a regular grid, constant coefficients, and the 2-norm. For ...Missing: original | Show results with:original
[4]
Computational Stability | CFD 101 | von Neuman Fourier analysis
The approach von Neumann used is based on two assumptions. First, that the difference equation can be linearized with respect to a small perturbation in the ...
[5]
von Neumann stability analysis - Richard Fitzpatrick
Our simple finite difference algorithm for solving the 1-d diffusion equation is subject to a numerical instability under certain circumstances.<|control11|><|separator|>
[6]
None
### Summary of Von Neumann Stability Analysis Historical Background
[7]
None
Below is a merged summary of Von Neumann's unpublished manuscripts from 1950 related to numerical integration of partial differential equations (PDEs) or stability analysis. The response consolidates all information from the provided segments into a dense, structured format using tables where appropriate, while retaining all details and avoiding redundancy. Since the system has a "no thinking token allowed" constraint, I will directly compile and organize the information as presented, ensuring completeness and clarity.
[8]
[PDF] On the Partial Difference Equations of Mathematical Physics
Problems involving the classical linear partial differential equations of mathematical physics can be reduced to algebraic ones of a very much simpler ...
[9]
[PDF] Stability of Numerical Schemes for PDE's. - MIT Mathematics
Stability simply means that the scheme does not amplify errors. Obviously this is very important, since errors are impossible to avoid in any numerical ...
[10]
[PDF] DAHLQUIST'S CLASSICAL PAPERS ON STABILITY THEORY
Sep 9, 2006 · Apparently, the numerical solution is of no use and, curiously, the solution be- comes better with the use of a wrong initial value (cases II): ...Missing: zero- | Show results with:zero-
[11]
[PDF] A special stability problem for linear multistep methods - Math-Unipd
The trapezoidal formula has the smallest truncation error among all linear multistep methods with a certain stability property. For this method error bounds.Missing: original | Show results with:original
[12]
[PDF] Survey of the Stability of Linear Finite Difference Equations
IX, 267–293 (1956). Survey of the Stability of Linear Finite Difference. Equations*. P. D. LAX and R. D. RICHTMYER. PART I. AN EQUIVALENCE THEOREM. 1 ...
[13]
[PDF] Numerical Methods for Partial Differential Equations - Seongjai Kim
Dec 11, 2023 · In solving PDEs numerically, the following are essential to consider: • physical laws governing the differential equations (physical understand-.
[14]
[PDF] Numerical Solution of Partial Differential Equations Finite Difference ...
It represents a decoupling of the odd and even time levels due to the nature of the scheme. Note that since there is no damping, this mode will persist in time.
[15]
[PDF] The Lax Equivalence Theorem
Theorem 4.6 (Lax Equivalence Theorem4). For a consistent finite difference scheme, stability is equivalent to convergence.
[16]
[PDF] Chapter 2: Finite Difference Approximations
Section 2.6 presents a von Neumann stability analysis of linear parabolic equations, including a study of time-explicit and time-implicit solution methods.
[17]
None
### Summary of Fourier Mode Decomposition in Von Neumann Stability Analysis
[18]
[PDF] Chapter 3. Finite Difference Methods for Hyperbolic Equations 1 ...
For a linear advection equation, we want the amplification factor to be 1, so that the wave does not grow or decay in time. The von Neumann stability analysis ...
[19]
[PDF] The Finite Difference Method - Stanford University
Application (in class): apply the von Neumann analysis to determine the stability of the FTFS scheme for the linear advection equation. 30 / 75. Page 40. AA214 ...
[20]
[PDF] PDE Stability Analysis - cs.Princeton
Von Neumann Stability Analysis. • Valid under assumptions (linear PDE, periodic ... Von Neumann: Advection Equation, FTCS. • Magnitude always ≥ 1 ...
[21]
[PDF] 1 Finite-Differences - UC Davis Climate and Global Change Group
Von Neumann analysis is a very powerful method for determining the stability properties of a numerical method. However, it can only be applied to problems on ...
[22]
[PDF] 17 Finite differences for the heat equation - UCSB Math
The stability analysis of these schemes showed that for stability of the first scheme we need. ∆t ≤ 1. 2. (∆x)2, while the second scheme is unconditionally ...Missing: Von | Show results with:Von
[23]
[PDF] 11. Finite Difference Methods for Partial Differential Equations
May 18, 2008 · Let us apply the von Neumann analysis to investigate the stability of the implicit scheme. Again, we need only look at the effect of the ...
[24]
[PDF] arXiv:2310.08274v1 [math.NA] 12 Oct 2023
Oct 12, 2023 · Von Neumann stability analysis, a well-known Fourier-based method, is a widely used technique for assess- ing stability in numerical ...
[25]
[PDF] ODE and PDE Stability Analysis - cs.Princeton
How does stability of method interact with stability of underlying. ODE? – ODE may prevent convergence (e.g., 𝛌 > 0). – Method may be unstable even when ODE is ...Missing: decomposition | Show results with:decomposition
[26]
[PDF] The Von Neumann Method for Stability Analysis
Jan 8, 2012 · The Von Neumann method offers an easy and simple way of assessing the stability properties of linear schemes with constant coefficients when the.
[27]
[PDF] Math/CS 714: Parabolic equations—von Neumann stability analysis ...
The von Neumann stability analysis is based on Fourier analysis and hence is generally limited to constant coefficient linear PDEs.
[28]
[PDF] FINITE DIFFERENCE AND SPECTRAL METHODS FOR ORDINARY ...
... von Neumann condition for vector or multistep formulas 183. 4.7. Stability of ... zero-stability" or sometimes \D-stability" for ODEs, and \Lax ...
[29]
[PDF] Computational Mechanics 6-1 David Apsley 6. METHODS FOR 2
A Von Neumann stability analysis is often used to assess the stability of numerical schemes for linear equations with constant coefficients. Basic idea: examine ...Missing: applicability | Show results with:applicability
[30]
[PDF] Exercises from Finite Difference Methods for Ordinary and Partial ...
Exercise 6.3 (consistency and zero-stability of LMMs) . ... (b) Using von Neumann analysis, show that this method is unconditionally ...<|control11|><|separator|>
[31]
[PDF] FROM SEMIDISCRETE TO FULLY DISCRETE - UMD MATH
In this context, we should highlight the following points com- paring our stability approach, with the classical von Neumann analysis which is part of the ...
[32]
von Neumann Stability Analysis of a Segregated Pressure-Based ...
The goal of this paper is to derive the von Neumann stability conditions for the pressure-based solution scheme, semi-implicit method for pressure-linked.<|control11|><|separator|>
[33]
On stability of numerical schemes via frozen coefficients and the ...
Aug 5, 2025 · The von Neumann stability analysis method is a common tool for assessing the stability of numerical methods such as finite difference ...
[34]
[PDF] Von Neumann stability analysis of Biot's general two-dimensional ...
The goal of this paper is to explore Von Neumann stability analysis of the GFEM for the. Biot problem using equal order interpolation elements (bilinear). While ...Missing: multidimensional | Show results with:multidimensional
[35]
von Neumann stability analysis of globally constraint-preserving ...
This paper catalogues the first step in the design of globally constraint-preserving DGTD schemes. The algorithms presented here are based on a novel DG-like ...
[36]
A Stability Analysis of Finite-Volume Advection Schemes Permitting ...
A one- and two-dimensional Von Neumann stability analysis of these finite-volume advection schemes is given.
[37]
[PDF] Stability analysis of finite-difference, pseudospectral and Fourier ...
Studying the properties of finite-difference methods can be carried out by the familiar von Neumann analysis, where the Fourier symbols of the difference scheme.
[38]
Chapter 10 Stability Analysis and Boundary Procedures
Stability analysis is done mainly by the von Neumann method and the energy method. The former has been widely used for linear (or linearized) equations, while ...Missing: limitations | Show results with:limitations
[39]
[PDF] The Matrix Method for Stability Analysis
Jan 10, 2010 · The Von Neumann method is based on the assumptions of the existence of a Fourier decomposition of the solution over the finite computational ...
[40]
[PDF] Stability Analysis of Numerical Boundary Conditions and Implicit ...
As mentioned previously, the von Neumann stability of the Cauchy problem is assured since the temporal integration scheme is assumed to be A-stable. The GKS ...