Allen-Cahn

anypinn create my-project --template allen-cahn

Stiff reaction-diffusion PDE with sharp interfaces. Forward problem demonstrating adaptive sampling and periodic boundary conditions.

Background

The Allen-Cahn equation, introduced by Allen and Cahn (1979), models phase separation and interface motion in binary alloys and other materials. The solution \(u\) represents a non-conserved order parameter that evolves under a double-well potential (\(u - u^3\) term), driving it toward the two stable phases \(u \approx \pm 1\). The diffusion coefficient \(\varepsilon\) controls the interface width; small \(\varepsilon\) produces sharp transition layers between phases, creating stiff dynamics that challenge standard PDE solvers and require adaptive sampling to resolve accurately.

Governing Equations

\[ \frac{\partial u}{\partial t} = \varepsilon\,\frac{\partial^2 u}{\partial x^2} + u - u^3 \]

where:

• \(u(x, t)\): order parameter (phase field)
• \(\varepsilon\): diffusion coefficient controlling interface width (known)

Default Configuration

The generated template uses the following values.

Known constants:

Symbol	Code constant	Value
\(\varepsilon\)	TRUE_EPSILON	\(0.01\)

Initial condition: \(u(x, 0) = -\tanh\!\left(\dfrac{x}{\sqrt{2\varepsilon}}\right)\), the stationary kink profile centered at \(x = 0\).

Boundary conditions: periodic (\(u(-1, t) = u(1, t)\)).

Domain: \(x \in [-1, 1], \quad t \in [0, 1]\)

Features Demonstrated

• AdaptiveSampler for resolving sharp interfaces
• Periodic boundary conditions
• Stiff PDE dynamics (small \(\varepsilon\))
• Forward problem (no parameter recovery)

Results