Partial differential equations solvers library

A library for numerical solution of partial differential equations (PDE) using finite difference method and Thomas algorithm

Features

Solution of parabolic equations of the form:

L(x,t,U)*∂U/∂t = ∂U( K(x,t,U)*∂U/∂x )/∂x + V(x,t,U)*∂U/∂x + F(x,t,U)

where U = U(x,t) is the unknown function (temperature, concentration, etc)

Solution of hyperbolic equations of the form:

M(x,t,U)*∂²U/∂t² = ∂U( K(x,t,U)*∂U/∂x )/∂x + V(x,t,U)*∂U/∂x + F(x,t,U)

where U = U(x,t) is the unknown function (displacement of string, etc)

Support for various boundary conditions:

• Dirichlet (function value at the boundary)
• Neumann (derivative value at the boundary)
• Robin (linear combination of function value and its derivative)

Efficient numerical methods:

• Finite difference method for derivative approximation
• Thomas algorithm for solving tridiagonal systems of linear equations

Prerequisites

• JDK 21
• Gradle (embedded in the project)

Building the project

./gradlew clean build

Generating documentation

./gradlew clean javadoc

Check build/doc/javadoc folder
Online documentation is available here

Supported equation types

Parabolic equations

ParabolicEquation.java

• Describes diffusion, heat conduction, and other dissipative processes
• Characterized by the presence of only first-order time derivative
• Solution is defined on the space-time domain [x1,x2]×[0,t2]

Hyperbolic equations

HyperbolicEquation.java

• Describes wave processes and oscillations
• Characterized by the presence of second-order time derivative
• Solution is defined on the space-time domain [x1,x2]×[0,t2]

Boundary conditions

Dirichlet (definite mode)

BorderCondition1.java

• Specifies function value at the boundary: U(x1,t) = g1(t) or U(x2,t) = g2(t)

Neumann (definite force)

BorderCondition2.java

• Specifies derivative value at the boundary: ∂U/∂x(x1,t) = g1(t) or ∂U/∂x(x2,t) = g2(t)

Robin (elastic fixing)

BorderCondition3.java

• Specifies linear combination of function value and its derivative: ∂U/∂x(x1,t) = h*(U(x1,t) - Theta(t)) or ∂U/∂x(x2,t) = h*(U(x2,t) - Theta(t))

Solution methods

Finite difference method

• Approximates derivatives using finite differences
• Transforms PDE into a system of linear equations
• Uses the Thomas algorithm to solve the resulting system

Thomas algorithm

• Efficient algorithm for solving tridiagonal systems of linear equations
• Time complexity O(n), where n is the system size

Library usage examples

Parabolic equations

• Diffusion problem with Dirichlet boundary conditions
• Diffusion problem with Neumann boundary conditions
• Heat transfer problem with mixed boundary conditions

Hyperbolic equations

• Wave equation (plucked string) with Dirichlet boundary conditions

Solution utilities

• Solution data saving and loading

Making new library release on GitHub

See instruction

Video with description of the project

License

MIT License