# 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 discontinuous 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 analysis functions will allow for simplifying the PDE before solving it, and constructing block symbolic functions like Jacobians.

## Constructors

`ModelingToolkit.PDESystem`

— Type`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

`systems`

systems: The internal systems. These are required to have unique names.

`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])
```

### 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.

## Boundary Conditions (WIP)

## Transformations

## Analyses

## 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.