📐 Von Neumann Stability Analysis
Compute stability criteria and amplification factors for finite difference schemes using Fourier analysis.
⚡ Fortran 90 Engine
Double Precision (IEEE 754)
✓ ISO / ASME Validated
Cfd
📊 Solver Telemetry
● ACTIVE
📥 Discretization Scheme & Courant Number
📖 Von Neumann Stability Condition:
$$u_j^n = V^n e^{i k j \Delta x} = V^n e^{i j \theta}$$
$$G(\theta) = \frac{V^{n+1}}{V^n}, \quad |G(\theta)| \le 1 \quad \forall\ \theta \in [0, \pi]$$
Current Scheme Criterion:
Conditionally Stable for 0 ≤ σ ≤ 1
Conditionally Stable for 0 ≤ σ ≤ 1
STABLE ✅
Stability Status
1.0000
Max Amplification ($\max |G|$)
0.0°
Critical Wave Angle ($\theta_{\text{crit}}$)
CFL=0.8
Discretization Parameter
💡
Numerical Stability Diagnostic
For 1st-Order Upwind (FOU) with Courant number $\sigma = 0.8$, the peak Fourier amplification factor is $\max |G(\theta)| = \mathbf{1.0000}$ at $\theta = 0.0^\circ$.
Because $|G(\theta)| \le 1$ across all wave modes, numerical errors remain bounded and damp out over time.
📈 Amplification Factor Spectrum: $|G(\theta)|$ vs Phase Angle $\theta$
Dashed Red: $|G| = 1.0$ (Stability Boundary)🔍 View Raw GNU Fortran Double-Precision Solver Output
SCHEME=Upwind 1st Order PARAM_LABEL=sigma (Courant) PARAM_VALUE= 8.00000000E-01 CRITERION=Stable if 0 <= sigma <= 1 DATA_START 0.00000000, 1.0000000000 0.03173326, 0.9999194435 0.06346652, 0.9996778163 0.09519978, 0.9992752450 0.12693304, 0.9987119405 0.15866630, 0.9979881985 0.19039955, 0.9971043993 0.22213281, 0.9960610081 0.25386607, 0.9948585751 0.28559933, 0.9934977361 0.31733259, 0.9919792123 0.34906585, 0.9903038113 0.38079911, 0.9884724268 0.41253237, 0.9864860396 0.44426563, 0.9843457179 0.47599889, 0.9820526175 0.50773215, 0.9796079832 0.53946541, 0.9770131484 0.57119866, 0.9742695369 0.60293192, 0.9713786626 0.63466518, 0.9683421309 0.66639844, 0.9651616395 0.69813170, 0.9618389791 0.72986496, 0.9583760347 0.76159822, 0.9547747861 0.79333148, 0.9510373095 0.82506474, 0.9471657784 0.85679800, 0.9431624647 0.88853126, 0.9390297405 0.92026451, 0.9347700786 0.95199777, 0.9303860549 0.98373103, 0.9258803489 1.01546429, 0.9212557461 1.04719755, 0.9165151390 1.07893081, 0.9116615293 1.11066407, 0.9066980291 1.14239733, 0.9016278635 1.17413059, 0.8964543715 1.20586385, 0.8911810091 1.23759711, 0.8858113503 1.26933037, 0.8803490899 1.30106362, 0.8747980455 1.33279688, 0.8691621594 1.36453014, 0.8634455013 1.39626340, 0.8576522704 1.42799666, 0.8517867974 1.45972992, 0.8458535476 1.49146318, 0.8398571225 1.52319644, 0.8338022626 1.55492970, 0.8276938495 1.58666296, 0.8215369082 1.61839622, 0.8153366096 1.65012947, 0.8090982720 1.68186273, 0.8028273638 1.71359599, 0.7965295046 1.74532925, 0.7902104676 1.77706251, 0.7838761804 1.80879577, 0.7775327264 1.84052903, 0.7711863456 1.87226229, 0.7648434349 1.90399555, 0.7585105482 1.93572881, 0.7521943958 1.96746207, 0.7459018433 1.99919533, 0.7396399096 2.03092858, 0.7334157647 2.06266184, 0.7272367263 2.09439510, 0.7211102551 2.12612836, 0.7150439500 2.15786162, 0.7090455412 2.18959488, 0.7031228832 2.22132814, 0.6972839451 2.25306140, 0.6915368006 2.28479466, 0.6858896159 2.31652792, 0.6803506363 2.34826118, 0.6749281709 2.37999443, 0.6696305756 2.41172769, 0.6644662340 2.44346095, 0.6594435368 2.47519421, 0.6545708592 2.50692747, 0.6498565362 2.53866073, 0.6453088361 2.57039399, 0.6409359324 2.60212725, 0.6367458738 2.63386051, 0.6327465522 2.66559377, 0.6289456705 2.69732703, 0.6253507078 2.72906028, 0.6219688848 2.76079354, 0.6188071278 2.79252680, 0.6158720333 2.82426006, 0.6131698315 2.85599332, 0.6107063521 2.88772658, 0.6084869888 2.91945984, 0.6065166677 2.95119310, 0.6047998155 2.98292636, 0.6033403315 3.01465962, 0.6021415613 3.04639288, 0.6012062747 3.07812614, 0.6005366463 3.10985939, 0.6001342403 3.14159265, 0.6000000000 DATA_END G_MAX= 1.0000000000 THETA_GMAX= 0.00000000 STABLE=YES
📘 Calculation Methodology: Von Neumann Fourier Spectral Stability
Mathematical Model & Theory
Von Neumann stability analysis expands numerical error into spatial Fourier harmonics $\epsilon_j^n = \hat{\epsilon}^n e^{i k x_j}$ and requires amplification factor $|G(\theta)| \le 1$ across all wavenumbers $\theta \in [-\pi, \pi]$:
$$|G(\theta)| \le 1 \quad \forall \theta \in [-\pi, \pi]$$
$$\text{1D Heat Equation: } G(\theta) = 1 - 2 Fo(1 - \cos\theta) \implies Fo \le 0.5$$
Assumptions
- Linear PDEs with constant coefficients.
- Periodic boundary conditions on uniform meshes.
Academic References
- Charney, J. G., & von Neumann, J. (1950): Tellus.
- Hirsch, C.: Numerical Computation, Wiley.
Worked Engineering Example
Problem Statement:
Calculate amplification factor $G(\pi)$ for 1D diffusion at $Fo = 0.4$ and $Fo = 0.6$.
Step-by-step Solution:
1. $G(\pi) = 1 - 4 Fo$.
2. At $Fo = 0.4$: $G = 1 - 1.6 = -0.6$ (Stable, $|G| \le 1$).
3. At $Fo = 0.6$: $G = 1 - 2.4 = -1.4$ (Unstable, $|G| > 1$).
Final Result:
Stable at $Fo = 0.4$ ($|G| = 0.6$), unstable at $Fo = 0.6$ ($|G| = 1.4$).
Calculate amplification factor $G(\pi)$ for 1D diffusion at $Fo = 0.4$ and $Fo = 0.6$.
Step-by-step Solution:
1. $G(\pi) = 1 - 4 Fo$.
2. At $Fo = 0.4$: $G = 1 - 1.6 = -0.6$ (Stable, $|G| \le 1$).
3. At $Fo = 0.6$: $G = 1 - 2.4 = -1.4$ (Unstable, $|G| > 1$).
Final Result:
Stable at $Fo = 0.4$ ($|G| = 0.6$), unstable at $Fo = 0.6$ ($|G| = 1.4$).