PDEBase.jl

PDEBase.jl provides common types and an interface for building discretizers of ModelingToolkit.jl PDESystems. It serves as a foundation for PDE discretization packages in the SciML ecosystem.

Installation

To install PDEBase.jl, use the Julia package manager:

using Pkg
Pkg.add("PDEBase")

Purpose

PDEBase.jl is designed to be extended by discretization packages, not used directly for solving PDEs. It provides:

Abstract types for discretizations, discrete spaces, and differential operators
Interface functions that discretization packages implement
Utility functions for symbolic manipulation of PDEs
Boundary condition handling with a structured type hierarchy

If you want to solve PDEs, use a package built on PDEBase:

MethodOfLines.jl - Finite difference discretization

Key Concepts

The PDESystem Interface

PDEBase defines how a ModelingToolkit.PDESystem is converted into a discretized system (typically an ODESystem). Key assumptions about the input PDESystem:

Well-defined domains with finite intervals for all variables
Consistent dependent variable signatures throughout
Boundary conditions specified at domain boundaries
Initial conditions for time-dependent problems

See Developer Extension API for complete details.

The Discretization Workflow

The symbolic_discretize function orchestrates the conversion:

Preprocessing - Handle complex numbers and array variables
Parsing - Create a VariableMap with all variable information
Boundary parsing - Structure boundary conditions by variable and dimension
Space construction - Build the discrete grid and variables
Equation processing - Convert each PDE to discrete equations
System generation - Combine into the output system

See Discretization Workflow for the complete pipeline.

Type Hierarchy

AbstractDiscretization (from SciMLBase)
├── AbstractEquationSystemDiscretization
└── AbstractOptimizationSystemDiscretization

AbstractDiscreteSpace
└── AbstractCartesianDiscreteSpace

AbstractBoundary
└── AbstractTruncatingBoundary
    ├── LowerBoundary
    ├── UpperBoundary
    └── AbstractInterfaceBoundary
        ├── InterfaceBoundary
        └── HigherOrderInterfaceBoundary

Documentation

Quick Reference

Abstract Types to Extend

Type	Purpose
`AbstractEquationSystemDiscretization`	For ODE/DAE-producing discretizations
`AbstractOptimizationSystemDiscretization`	For optimization problems
`AbstractDiscreteSpace`	Discrete spatial representation
`AbstractDifferentialDiscretizer`	Differential operator handling

Key Interface Functions

Function	Purpose
`construct_discrete_space`	Build the discrete grid and variables
`construct_differential_discretizer`	Prepare derivative approximations
`discretize_equation!`	Convert a PDE to discrete equations
`generate_system`	Create the final symbolic system

Exported Utility Functions

Category	Functions
Differential analysis	`count_differentials`, `differential_order`, `has_derivatives`, `find_derivative`, `d_orders`
Expression manipulation	`get_depvars`, `split_terms`, `split_additive_terms`, `subs_alleqs!`
Variable access	`depvars`, `indvars`, `all_ivs`, `ivs`, `depvar`, `x2i`
Boundary utilities	`isupper`, `isinterface`, `isperiodic`, `filter_interfaces`, `haslowerupper`, `has_interfaces`

ModelingToolkit.jl - Defines the PDESystem type
MethodOfLines.jl - Finite difference discretization built on PDEBase
DifferentialEquations.jl - Solves the resulting ODE/DAE systems

Contributing

Contributions are welcome! Please open an issue or pull request on GitHub.