Class FiniteDifference

java.lang.Object

org.episteme.core.mathematics.analysis.pde.FiniteDifference

public class FiniteDifference
extends Object

Finite Difference Methods for solving Partial Differential Equations.

Discretizes PDEs on grid, approximates derivatives with differences. Handles: heat equation, wave equation, Poisson equation.

Since:

1.0

Author:

Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

Constructor Summary

Constructors

Constructor	Description
FiniteDifference()

Method Summary

Modifier and Type	Method	Description
static Real[][]	heatEquation1D(Function<Real,Real> initial, Real alpha, Real L, Real T, int nx, int nt)	Solves 1D heat equation: ∂u/∂t = α ∂²u/∂x²
static Real[][]	poisson2D(Function<Real[],Real> source, int nx, int ny, Real lx, Real ly, int maxIter, Real tolerance)	Solves 2D Poisson equation: ∇²u = f on rectangular domain.
static Real[][]	waveEquation1D(Function<Real,Real> initial, Function<Real,Real> initialVelocity, Real c, Real L, Real T, int nx, int nt)	Solves 1D wave equation: ∂²u/∂t² = c² ∂²u/∂x²

Methods inherited from class Object

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

Constructor Details

FiniteDifference

public FiniteDifference()

Method Details

heatEquation1D

public static Real[][] heatEquation1D(Function<Real,Real> initial,
 Real alpha,
 Real L,
 Real T,
 int nx,
 int nt)

Solves 1D heat equation: ∂u/∂t = α ∂²u/∂x²

Forward Euler (explicit) time stepping. Boundary conditions: u(0,t) = u(L,t) = 0

Parameters:

initial - initial condition u(x,0)

alpha - thermal diffusivity

L - domain length

T - final time

nx - number of spatial points

nt - number of time steps

Returns:

solution u[time][space]

waveEquation1D

public static Real[][] waveEquation1D(Function<Real,Real> initial,
 Function<Real,Real> initialVelocity,
 Real c,
 Real L,
 Real T,
 int nx,
 int nt)

Solves 1D wave equation: ∂²u/∂t² = c² ∂²u/∂x²

Explicit central differences in space and time.

Parameters:

initial - initial displacement u(x,0)

initialVelocity - initial velocity ∂u/∂t(x,0)

c - wave speed

L - domain length

T - final time

nx - spatial points

nt - time steps

Returns:

solution u[time][space]

poisson2D

public static Real[][] poisson2D(Function<Real[],Real> source,
 int nx,
 int ny,
 Real lx,
 Real ly,
 int maxIter,
 Real tolerance)

Solves 2D Poisson equation: ∇²u = f on rectangular domain.

Uses Jacobi iteration (simple iterative solver). ∂²u/∂x² + ∂²u/∂y² = f

Parameters:

source - source term f(x,y)

nx - , ny grid points in x, y

lx - , ly domain size

maxIter - max iterations

tolerance - convergence tolerance

Returns:

solution u[x][y]