# PDESystem

PDESystem is the common symbolic PDE specification for the SciML ecosystem. It is currently being built as a component of the ModelingToolkit ecosystem,

## Vision

The vision for the common PDE interface is that a user should only have to specify their PDE once, mathematically, and have instant access to everything as simple as a finite difference method with constant grid spacing, to something as complex as a distributed multi-GPU discrete Galerkin method.

The key to the common PDE interface is a separation of the symbolic handling from the numerical world. All of the discretizers should not "solve" the PDE, but instead be a conversion of the mathematical specification to a numerical problem. Preferably, the transformation should be to another ModelingToolkit.jl AbstractSystem, but in some cases this cannot be done or will not be performant, so a SciMLProblem is the other choice.

These elementary problems, such as solving linear systems Ax=b, solving nonlinear systems f(x)=0, ODEs, etc. are all defined by SciMLBase.jl, which then numerical solvers can all target these common forms. Thus someone who works on linear solvers doesn't necessarily need to be working on a Discontinuous Galerkin or finite element library, but instead "linear solvers that are good for matrices A with properties ..." which are then accessible by every other discretization method in the common PDE interface.

Similar to the rest of the AbstractSystem types, transformation and analyses functions will allow for simplifying the PDE before solving it, and constructing block symbolic functions like Jacobians.

## Constructors

ModelingToolkit.PDESystemType
struct PDESystem <: AbstractMultivariateSystem

A system of partial differential equations.

Fields

• eqs

The equations which define the PDE

• bcs

The boundary conditions

• domain

The domain for the independent variables.

• ivs

The independent variables

• dvs

The dependent variables

• ps

The parameters

• defaults

defaults: The default values to use when initial conditions and/or parameters are not supplied in ODEProblem.

• connector_type

type: type of the system

• name

name: the name of the system

Example

using ModelingToolkit

@parameters x
@variables t u(..)
Dxx = Differential(x)^2
Dtt = Differential(t)^2
Dt = Differential(t)

#2D PDE
C=1
eq  = Dtt(u(t,x)) ~ C^2*Dxx(u(t,x))

# Initial and boundary conditions
bcs = [u(t,0) ~ 0.,# for all t > 0
u(t,1) ~ 0.,# for all t > 0
u(0,x) ~ x*(1. - x), #for all 0 < x < 1
Dt(u(0,x)) ~ 0. ] #for all  0 < x < 1]

# Space and time domains
domains = [t ∈ (0.0,1.0),
x ∈ (0.0,1.0)]

@named pde_system = PDESystem(eq,bcs,domains,[t,x],[u])
source

### Domains (WIP)

Domains are specifying by saying indepvar in domain, where indepvar is a single or a collection of independent variables, and domain is the chosen domain type. A 2-tuple can be used to indicate an Interval. Thus forms for the indepvar can be like:

t ∈ (0.0,1.0)
(t,x) ∈ UnitDisk()
[v,w,x,y,z] ∈ VectorUnitBall(5)

#### Domain Types (WIP)

• Interval(a,b): Defines the domain of an interval from a to b (requires explicit

import from DomainSets.jl, but a 2-tuple can be used instead)

## discretize and symbolic_discretize

The only functions which act on a PDESystem are the following:

• discretize(sys,discretizer): produces the outputted AbstractSystem or SciMLProblem.
• symbolic_discretize(sys,discretizer): produces a debugging symbolic description of the discretized problem.

## Discretizer Ecosystem

### NeuralPDE.jl: PhysicsInformedNN

NeuralPDE.jl defines the PhysicsInformedNN discretizer which uses a DiffEqFlux.jl neural network to solve the differential equation.

### DiffEqOperators.jl: MOLFiniteDifference (WIP)

DiffEqOperators.jl defines the MOLFiniteDifference discretizer which performs a finite difference discretization using the DiffEqOperators.jl stencils. These stencils make use of NNLib.jl for fast operations on semi-linear domains.