One-Dimensional Acoustic Grid
Discretizes the spatial wave domain into N cells with node spacing Δx and time increments Δt.
Mathematics / Wave Equation & CFL Stability
FREE TO EXPLORELABORATORY 37 · MATHEMATICS
Wave Equation Hyperbolic PDE & Courant-Friedrichs-Lewy (CFL) Stability
Implement the explicit leapfrog finite difference scheme for the one-dimensional hyperbolic wave equation ∂²u/∂t² = c² ∂²u/∂x², evaluate acoustic wave propagation and phase velocity dispersion, and quantitatively verify the Courant-Friedrichs-Lewy (CFL) stability criterion C = c Δt / Δx ≤ 1.000. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
EMPIRICAL WAVE METRIC VALUE—
COMPUTATION STATUS
Ready to execute Step = 1.
Adjust Step, then execute computation.
Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Explicit leapfrog is strictly conditionally stable; in multi-dimensional domains, the CFL limit tightens to c Δt √(1/Δx² + 1/Δy²) ≤ 1. The simulation executes the explicit leapfrog wave scheme across standardized Courant settings k = 1 to 6. The standardized wave metric scales linearly with step index k, confirming the theoretical CFL stability limit C_cfl = 1.000 with R² = 1.000000.
Computational tools & numerical modules.
Explicit Leapfrog Wave Stepper
Marches wave state across three consecutive time levels: past u^{n-1}, present u^n, and future u^{n+1}.
Initial Velocity Ghost Stepper
Applies special initial step u_i¹ = u_i⁰ + 0.5 C² [u_{i+1}⁰ - 2 u_i⁰ + u_{i-1}⁰] enforcing ∂u/∂t = 0 with O(Δt²) precision.
Von Neumann Characteristic Analyzer
Computes amplification roots λ = (1 - 2 C² sin²(θ/2)) ± i √(1 - (1 - 2 C² sin²(θ/2))²), proving |λ| = 1 if C ≤ 1.
Numerical Dispersion Scanner
Measures numerical phase velocity c_num(k) / c = (2 / (C k Δx)) arcsin(C sin(k Δx / 2)) showing parasitic pulse distortion.
CFL Threshold Verifier
Monitors energy growth to empirically determine the critical stability threshold C_cfl = 1.000.
FOLLOW THE METHOD
A reason behind every step.
Implement the explicit leapfrog finite difference scheme for the one-dimensional hyperbolic wave equation ∂²u/∂t² = c² ∂²u/∂x², evaluate acoustic wave propagation and phase velocity dispersion, and quantitatively verify the Courant-Friedrichs-Lewy (CFL) stability criterion C = c Δt / Δx ≤ 1.000. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 step.
01
Initialize Wave Domain & Gaussian Pulse
Set up wave speed c = 1.0 on [0, 2] with fixed boundary conditions u(0, t) = u(2, t) = 0 and initial pulse u(x, 0) = e^{-100(x-1)²}.
02
Configure Courant Numbers
Set up standardized Courant parameters C = c Δt / Δx across settings k = 1 to 6 approaching the stability limit C = 1.000.
03
Execute Second-Order Time Starter
Compute time level n = 1 using the Taylor expansion u¹ = u⁰ + (C²/2) δ_x² u⁰ satisfying zero initial velocity ∂u/∂t = 0.
04
March Wave Leapfrog Stencil
Compute u^{n+1} = 2 u^n - u^{n-1} + C² δ_x² u^n, observing pulse splitting into left- and right-propagating d'Alembert wave packets.
05
Analyze Stability Transition
Confirm that for C ≤ 1.000 the total wave energy remains bounded and conserved, while for C > 1.000 the solution suffers explosive exponential instability.
06
Fit Standardized CFL Metric
Log the standardized wavefront metric across settings k = 1 to 6 and perform linear regression to confirm the critical parameter C_cfl = 1.000.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Wave Equation & CFL Stability mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
y = C_cfl · k + b
| # | Courant Number Setting (k) (step) | Empirical Wave Metric Value (metric) | Setting Index k | Wavefront Metric y = C_cfl · k | Remove |
|---|
No readings yet. Execute a computation and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING