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| { | ||
| "header": { | ||
| "name": "Advection 1d Model", | ||
| "schema": "https://raw.githubusercontent.com/DARPA-ASKEM/Model-Representations/petrinet_v0.1/petrinet/petrinet_schema.json", | ||
| "schema_name": "petrinet", | ||
| "description": "Advection 1d as Petrinet model created by Drisana", | ||
| "model_version": "0.1" | ||
| }, | ||
| "model": { | ||
| "states": [ | ||
| { | ||
| "id": "u_0", | ||
| "name": "u_0", | ||
| "description": "density" | ||
| }, | ||
| { | ||
| "id": "u_1", | ||
| "name": "u_1", | ||
| "description": "density" | ||
| }, | ||
| { | ||
| "id": "u_2", | ||
| "name": "u_2", | ||
| "description": "density" | ||
| }, | ||
| { | ||
| "id": "u_3", | ||
| "name": "u_3", | ||
| "description": "density" | ||
| }, | ||
| { | ||
| "id": "u_4", | ||
| "name": "u_4", | ||
| "description": "density" | ||
| } | ||
| ], | ||
| "transitions": [ | ||
| { | ||
| "id": "t_0", | ||
| "input": [ | ||
| "u_0" | ||
| ], | ||
| "output": [ | ||
| "u_1" | ||
| ], | ||
| "properties": { | ||
| "name": "t_0" | ||
| } | ||
| }, | ||
| { | ||
| "id": "t_1", | ||
| "input": [ | ||
| "u_1" | ||
| ], | ||
| "output": [ | ||
| "u_2" | ||
| ], | ||
| "properties": { | ||
| "name": "t_1" | ||
| } | ||
| }, | ||
| { | ||
| "id": "t_2", | ||
| "input": [ | ||
| "u_2" | ||
| ], | ||
| "output": [ | ||
| "u_3" | ||
| ], | ||
| "properties": { | ||
| "name": "t_2" | ||
| } | ||
| }, | ||
| { | ||
| "id": "t_3", | ||
| "input": [ | ||
| "u_3" | ||
| ], | ||
| "output": [ | ||
| "u_4" | ||
| ], | ||
| "properties": { | ||
| "name": "t_3" | ||
| } | ||
| } | ||
| ] | ||
| }, | ||
| "semantics": { | ||
| "ode": { | ||
| "rates": [ | ||
| { | ||
| "target": "t_0", | ||
| "expression": "(a/dx)*(u_1-u_0)" | ||
| }, | ||
| { | ||
| "target": "t_1", | ||
| "expression": "(a/dx)*(u_2-u_1)" | ||
| }, | ||
| { | ||
| "target": "t_2", | ||
| "expression": "(a/dx)*(u_3-u_2)" | ||
| }, | ||
| { | ||
| "target": "t_3", | ||
| "expression": "(a/dx)*(u_4-u_3)" | ||
| } | ||
| ], | ||
| "initials": [ | ||
| { | ||
| "target": "u_0", | ||
| "expression": "0.1" | ||
| }, | ||
| { | ||
| "target": "u_1", | ||
| "expression": "0.5" | ||
| }, | ||
| { | ||
| "target": "u_2", | ||
| "expression": "1.0" | ||
| }, | ||
| { | ||
| "target": "u_3", | ||
| "expression": "0.5" | ||
| }, | ||
| { | ||
| "target": "u_4", | ||
| "expression": "0.1" | ||
| } | ||
| ], | ||
| "parameters": [ | ||
| { | ||
| "id": "a", | ||
| "value": 1.0, | ||
| "distribution": { | ||
| "type": "StandardUniform1", | ||
| "parameters": { | ||
| "minimum": 0.0, | ||
| "maximum": 1.0 | ||
| } | ||
| } | ||
| }, | ||
| { | ||
| "id": "dx", | ||
| "value": 1.0, | ||
| "distribution": { | ||
| "type": "StandardUniform1", | ||
| "parameters": { | ||
| "minimum": 0.0, | ||
| "maximum": 1.0 | ||
| } | ||
| } | ||
| } | ||
| ], | ||
| "time": { | ||
| "id": "t", | ||
| "units": { | ||
| "expression": "day", | ||
| "expression_mathml": "<ci>day</ci>" | ||
| } | ||
| } | ||
| } | ||
| } | ||
| } |
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| { | ||
| "structure_parameters": [ | ||
| { | ||
| "name": "schedules", | ||
| "schedules": [ | ||
| { | ||
| "timepoints": [ | ||
| 0, | ||
| 1, | ||
| 2, | ||
| 3, | ||
| 4, | ||
| 5, | ||
| 6, | ||
| 7, | ||
| 8, | ||
| 9, | ||
| 10 | ||
| ] | ||
| } | ||
| ] | ||
| } | ||
| ], | ||
| "config": { | ||
| "use_compartmental_constraints": false, | ||
| "normalization_constant": 1.0, | ||
| "tolerance": 1e-1, | ||
| "verbosity": 20, | ||
| "dreal_mcts": true, | ||
| "save_smtlib": "./out", | ||
| "substitute_subformulas": false, | ||
| "series_approximation_threshold": null, | ||
| "dreal_log_level": "none", | ||
| "profile": false | ||
| } | ||
| } |
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| from sympy import Derivative, E, Eq, Function, pde_separate, symbols | ||
| from sympy.parsing.latex import parse_latex | ||
|
|
||
| ### Symbol initialization | ||
| x, y, z, t, i, n, dx, dt, a = symbols("x y z t i n dx dt a") | ||
| init_printing(use_unicode=True) | ||
|
|
||
| u = symbols("u", cls=Function) | ||
| u = Function("u") | ||
| dudx = u(x, t).diff(x) | ||
| dudt = u(x, t).diff(t) | ||
|
|
||
| ### Solve for discrete derivatives - other arbitrary schemes for discretizing time and space can be brought in here. This is general, so these can be substituted into other PDEs. | ||
|
|
||
| backward_space_derivative = dudx.as_finite_difference( | ||
| [dx * (i - 1), dx * i] | ||
| ).subs(t, dt * i) | ||
| forward_space_derivative = dudx.as_finite_difference( | ||
| [dx * (i), dx * (i + 1)] | ||
| ).subs(t, dt * i) | ||
| forward_time_derivative = dudt.as_finite_difference( | ||
| [dt * (n), dt * (n + 1)] | ||
| ).subs(x, dx * i) | ||
|
|
||
| ### Parse advection equation. This one is from LaTeX but can use other sources too. | ||
|
|
||
| advection_parse = parse_latex( | ||
| r"\frac{\partial q}{\partial t} + a\frac{\partial q}{\partial x} = 0" | ||
| ) | ||
|
|
||
|
|
||
| ### Substitute into parsed equation - input definitions for u and discrete derivatives | ||
|
|
||
| advection_parse = advection_parse.subs(Symbol("q"), u(x, t)) | ||
|
|
||
| advection_parse = advection_parse.subs( | ||
| Derivative(u(x, t), t), forward_time_derivative | ||
| ) | ||
| advection_parse = advection_parse.subs( | ||
| Derivative(u(x, t), x), backward_space_derivative | ||
| ) | ||
|
|
||
| print( | ||
| advection_parse | ||
| ) ### Advection equation in terms of discrete time and space points | ||
|
|
||
| ### Solve for u at next timepoint ("n+1") in terms of previous points | ||
|
|
||
| next_time_step_advection_parse = solve( | ||
| advection_parse, u(dx * i, dt * (n + 1)) | ||
| ) | ||
|
|
||
| print( | ||
| next_time_step_advection_parse | ||
| ) ## matches FTBS on upwind scheme page of https://indico.ictp.it/event/a06220/session/18/contribution/10/material/0/2.pdf for a > 0 |