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How to Interpret this Dashboard

1. The Big Picture: What is this Simulator?

This simulator models a **1D Reaction-Diffusion Partial Differential Equation (PDE)**. It describes how a quantity, $u(x,t)$, changes over time ($t$) and space ($x$) due to two main processes: diffusion (spreading out) and local reactions (growth or decay).

The governing equation is: $$ \frac{\partial u}{\partial t} = \nu_{eff} \frac{\partial^2 u}{\partial x^2} + \beta_{NL} u(1-u) - \gamma_{diss} u^3 $$

2. Simulation Parameters Explained

• System Length (L) & Grid Points (N): Define the 1D space.
• Total Time (T): How long the simulation runs.
• Diff. Coefficient ($\nu_{eff}$): Controls diffusion, the tendency for $u$ to smooth out. Analogous to viscosity.
• Non-linear Growth ($\beta_{NL}$): The "reaction" part. It makes $u$ grow, but the $u(1-u)$ term limits this growth.
• Cubic Damping ($\gamma_{diss}$): An additional dissipation term that removes "energy" from the system, often leading to stabilization.
• Initial Condition: The starting shape of $u$ at time $t=0$. The "Custom Formula" option allows you to enter any JavaScript expression using variables `x`, `L`, and `N`, and the `Math` object.
• Boundary Condition: Determines behavior at the domain edges. 'Periodic' wraps around (like a circle), while 'Dirichlet' fixes the edges to zero (zero-flux).

3. Numerical Method: 4th-Order Runge-Kutta (RK4)

This simulator uses the **4th-Order Runge-Kutta (RK4)** method to solve the PDE over time. This is a highly accurate and stable numerical scheme, representing a significant improvement over simpler methods like Forward Euler. It works by taking four intermediate "test steps" within each time interval to calculate a weighted average slope, which dramatically reduces numerical error and allows for more reliable simulations, especially for complex or chaotic systems.

4. Time Step (dt) & Numerical Stability

The Time Step (dt) is one of the most critical parameters you can control. It determines the size of the discrete time intervals the RK4 solver uses.

• Small dt: Leads to a more accurate and stable simulation, but requires more computational steps and thus takes longer to run.
• Large dt: The simulation runs faster, but if `dt` is too large, the numerical solution can become unstable and "blow up", producing non-physical results (`Infinity` or `NaN`).

To help you, the dashboard includes a real-time Stability Indicator (CFL). This value is based on the Courant-Friedrichs-Lewy condition for the diffusion part of the equation. As a rule of thumb:

5. Simulation Presets Explained

The presets are designed to quickly load parameter configurations that demonstrate interesting and distinct physical phenomena.

• Pure Diffusion: Isolates the diffusion term ($\nu_{eff}$). This shows the system behaving like a heat equation, where any initial shape smooths out over time and decays towards a uniform state.
• Growth & Decay: Focuses on the non-linear reaction terms. This preset shows how the field $u$ can grow locally due to the $\beta_{NL}$ term and then become limited or stabilized by the $\gamma_{diss}$ damping term.
• Propagating Wave: Demonstrates a balance between diffusion and reaction that results in a traveling wave front. A step-like initial condition moves across the domain with a relatively stable shape.
• Pattern Formation: A configuration where the interplay between diffusion and reaction leads to the emergence of complex, stable, and repeating spatial patterns from a random initial state. This is analogous to Turing patterns.
• Turbulent-like Chaos: With high growth and low diffusion, the system enters a complex, chaotic state. The power spectrum for this state is typically broad, indicating energy across many spatial scales, analogous to the energy cascade in turbulence.
• Shockwave Analogue: This preset uses a step initial condition with parameters that cause the front to steepen as it propagates, forming a sharp gradient that is a 1D analogue of a shockwave.
• Smooth Decay (Global Regularity Analogue): Starts with a random, noisy state, but high diffusion ($\nu_{eff}$) dominates all other terms, quickly smoothing the solution and causing it to decay. This demonstrates a "globally regular" solution that remains well-behaved for all time, which is one possible outcome for the Navier-Stokes equations.
• Wave Steepening (Shock Formation): Begins with a smooth sine wave. Very low diffusion allows the non-linear growth term to "push" the wave forward, causing its front to become progressively steeper until it forms a near-vertical shock. This shows how sharp gradients can emerge from smooth initial data.
• Intermittency (Turbulent Puffs): Creates a state analogous to a key feature of real-world turbulence. The simulation shows long periods of calm or simple behavior, punctuated by sharp, chaotic bursts of activity. This is reflected in a high Kurtosis value in the metrics.
• Singularity Analogue (Finite-Time Blow-up): This is an educational preset with an intentionally unstable time-step (dt). Non-linear growth overwhelms the simulation's ability to remain stable, causing the values to rapidly shoot towards infinity. This visually demonstrates the concept of a "finite-time blow-up" or "singularity," the very phenomenon the Navier-Stokes conjecture questions the existence of.

6. How to Analyze the Plots & Data

After running a simulation, the "Detailed Analysis" section provides powerful tools:

Term Contributions Plot

Found below the main playback controls, this plot shows the individual components of the PDE in real-time. You can see how the Diffusion (blue), Reaction (green), and total Change (du/dt) (red dashed) contribute to the evolution of $u$. This is key to understanding why $u$ is changing at specific locations.

Final State Summary & AI Analysis

After a successful run, this new section appears. It provides key quantitative statistics about the final state, such as dominant pattern size (wavelength), correlation scales, and distribution shape (skewness, kurtosis). You can click Generate AI Analysis to send these stats to the Gemini LLM for an expert-level interpretation of the result.

Distribution Histograms

These plots show the statistical distribution of key quantities at the final simulation time. They reveal the overall character of the solution (e.g., is it mostly zero with a few peaks, or is it highly varied?).

Energy Spectrum Analysis

This tab shows how the system's "energy" ($u^2$) is distributed across different spatial frequencies (wavenumbers, $k$). It's a key tool for analyzing patterns. A sharp peak at a specific $k$ indicates a dominant wavelength. A broad spectrum suggests chaotic or noisy behavior. The live plot shows this evolving, and the Energy Spectrogram (k-t diagram) shows the complete history after the simulation is finished.

Phase Space Portrait

This plot visualizes the system's state by plotting the field value ($u$) against its spatial derivative ($\partial u / \partial x$). The shape of the resulting curve provides deep insights. For example, a simple point indicates a uniform state, a closed loop can indicate a periodic (wave-like) solution, and more complex shapes can suggest chaotic dynamics or the presence of "attractors."

Time-Space Plot (Hovmöller Diagram)

This visualization provides a complete history of the simulation. It plots space ($x$) horizontally and time ($t$) vertically (flowing downwards). The color at each point shows the value of $u(x,t)$, making it easy to see how patterns travel, grow, or stabilize over the entire simulation run.

Correlation Analysis

These advanced plots provide a quantitative look at the structural properties of the simulation's output.

• Spatial Autocorrelation: This function, $C(r)$, measures how similar the field $u(x)$ is to a shifted version of itself, $u(x+r)$.
  • A peak at a lag distance $r > 0$ indicates a dominant wavelength or pattern size in the data. The location of the first peak is a good measure of the characteristic length scale.
  • A rapid decay to zero implies a noisy or random spatial structure with no repeating patterns.
• Temporal Autocorrelation: This function, $C(\tau)$, measures the "memory" of the system by comparing its spatial state at time $t$ to its state at a later time $t+\tau$.
  • A slow decay indicates that the system is highly predictable and its future state is strongly related to its past.
  • A rapid decay to zero (a short correlation time) is a hallmark of chaotic or unpredictable behavior, where small initial differences grow exponentially fast.

Interactive Live Plot

Hover your mouse over the main "Live Simulation" plot. A tooltip will appear, showing you the precise values of $u$ and its first and second derivatives at that specific point in space. This is useful for inspecting sharp gradients or other local features.

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Legal Disclaimers & Terms of Use

1. Experimental Research Tool

This 1D Fluid Dynamics Analogue Simulator (the "Tool") is provided by Quantum Q PTE. LTD. ("Quantum Q") as an experimental research and educational tool, currently in a beta development phase. It is intended solely for use by the scientific, mathematical, and engineering communities for conceptual exploration and educational purposes. The Tool is not designed, certified, or intended for use in any commercial, industrial, clinical, or mission-critical applications.

2. No Warranty

THE TOOL IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, AND NONINFRINGEMENT. IN NO EVENT SHALL QUANTUM Q, ITS DIRECTORS, EMPLOYEES, OR AFFILIATES BE LIABLE FOR ANY CLAIM, DAMAGES, OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT, OR OTHERWISE, ARISING FROM, OUT OF, OR IN CONNECTION WITH THE TOOL OR THE USE OR OTHER DEALINGS IN THE TOOL.

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You expressly understand and agree that Quantum Q, its directors, and its employees shall not be liable for any direct, indirect, incidental, special, consequential, or exemplary damages, including but not to, damages for loss of profits, goodwill, use, data, or other intangible losses (even if Quantum Q has been advised of the possibility of such damages), resulting from the use or the inability to use the Tool. The entire risk as to the quality, performance, and accuracy of the Tool is with you. This includes, but is not limited to, any damages or injury caused by any failure of performance, error, omission, interruption, defect, or bug within the software, particularly given its acknowledged experimental and beta status. Should the Tool prove defective, you assume the entire cost of all necessary servicing, repair, or correction, and you agree to indemnify and hold harmless Quantum Q and its employees from any and all claims arising from your use of the Tool.

4. No Guarantee of Accuracy

While efforts have been made to ensure the mathematical and numerical integrity of the simulation, Quantum Q makes no guarantee as to the accuracy, reliability, or completeness of the results generated by the Tool. The outputs are for illustrative and conceptual purposes only and should not be used for peer-reviewed publications, engineering design, or any other formal application without independent verification and validation. As stated prominently within the Tool, this simulator is an *analogue* and is not a direct simulation of the 3D Navier-Stokes equations.

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While the primary governing law is that of the Republic of Singapore, these terms are intended to be broadly applicable to users globally. For users in jurisdictions where English law is applicable or preferred, these terms shall be interpreted in a manner consistent with the general principles of English contract law, to the extent that such interpretation does not conflict with the mandatory laws of Singapore. The parties acknowledge and agree that the principles of common law and equity, as applied in English law, may be considered in interpreting these terms where Singapore law is silent or where such principles provide additional clarity, provided always that Singapore law remains the ultimate governing law.

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