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 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
— Typestruct 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
: The default values to use when initial conditions and/or parameters are not supplied inODEProblem
.
connector_type
: Type of the system.
systems
: The internal systems. These are required to have unique names.
analytic
: A vector of explicit symbolic expressions for the analytic solutions of each dependent variable. e.g.analytic = [u(t, x) ~ a*sin(c*t) * cos(k*x)]
.
analytic_func
: A vector of functions for the analytic solutions of each dependent variable. Will be generated fromanalytic
if not provided. Should have the same argument signature as the variable, and aps
argument as the last argument, which takes an indexable of parameter values in the order you specified them inps
. e.g.analytic_func = [u(t, x) => (ps, t, x) -> ps[1]*sin(ps[2]*t) * cos(ps[3]*x)]
.
name
: The name of the system.
metadata
: Metadata for the system, to be used by downstream packages.
gui_metadata
: Metadata for MTK GUI.
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 froma
tob
(requires explicit import fromDomainSets.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 outputtedAbstractSystem
orSciMLProblem
.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.
MethodOfLines.jl: MOLFiniteDifference
MethodOfLines.jl defines the MOLFiniteDifference
discretizer which performs a finite difference discretization. Includes support for higher approximation order stencils and nonuniform grids.