Class Laplacian

java.lang.Object

org.episteme.core.mathematics.analysis.vectorcalculus.Laplacian

public class Laplacian
extends Object

Computes the Laplacian of a scalar field.

The Laplacian is the divergence of the gradient: ∇²f = ∇·(∇f)

Definition: ∇²f = ∂²f/∂x₁² + ∂²f/∂x₂² + ... + ∂²f/∂xₙ²

Physical interpretation and applications: - Heat equation: ∂T/∂t = α∇²T (heat diffusion) - Wave equation: ∂²u/∂t² = c²∇²u (wave propagation) - Laplace's equation: ∇²φ= 0 (electrostatics, gravity) - Poisson's equation: ∇²φ= -ρ/ε₀ (electrostatics with charges) - Schrödinger equation: iℏ∂ψ/∂t = -ℏ²/(2m)∇²ψ + Vψ - Harmonic functions: functions where ∇²f = 0

Since:

1.0

Author:

Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

Constructor Summary

Constructors

Constructor	Description
Laplacian()

Method Summary

Modifier and Type	Method	Description
static ScalarField<PointND>	asField(ScalarField<PointND> field, Real h)	Returns a scalar field representing the Laplacian of the input field.
static Real	compute(ScalarField<PointND> field, PointND point, Real h)	Computes the Laplacian of a scalar field at a point.
static Vector<Real>	computeVector(VectorField<PointND> field, PointND point, Real h)	Computes the vector Laplacian of a vector field.

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

Laplacian

public Laplacian()

Method Details

compute

public static Real compute(ScalarField<PointND> field,
 PointND point,
 Real h)

Computes the Laplacian of a scalar field at a point.

Uses second-order central difference: ∂²f/∂xᵢ² ≈ (f(x + h*eᵢ) - 2f(x) + f(x - h*eᵢ)) / h²

Parameters:

field - the scalar field

point - the point at which to compute Laplacian

h - the step size for numerical differentiation

Returns:

the Laplacian (scalar value)

asField

public static ScalarField<PointND> asField(ScalarField<PointND> field,
 Real h)

Returns a scalar field representing the Laplacian of the input field.

Parameters:

field - the scalar field

h - the step size for numerical differentiation

Returns:

the Laplacian as a scalar field

computeVector

public static Vector<Real> computeVector(VectorField<PointND> field,
 PointND point,
 Real h)

Computes the vector Laplacian of a vector field.

For a vector field F, the vector Laplacian is: ∇²F = (∇²F₁, ∇²F₂, ..., ∇²Fₙ)

Note: In some contexts, ∇²F = ∇(∇·F) - ∇×(∇×F), but this is only equivalent in 3D and for certain coordinate systems.

Parameters:

field - the vector field

point - the point

h - the step size

Returns:

the vector Laplacian