LaTeX to state-update equations from SKEMA meeting

scratch/sympy_pde_example.py

@@ -0,0 +1,55 @@
+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