# How to get PDE solutions when doing foward time steps

**URL:** <https://discourse.julialang.org/t/how-to-get-pde-solutions-when-doing-foward-time-steps/105419>\
**Category:** Modelling & Simulations\
**Created:** [October 26, 2023, 5:58am UTC](https://discourse.julialang.org/t/how-to-get-pde-solutions-when-doing-foward-time-steps/105419 "2023-10-26T05:58:58Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![aplund](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/aplund/32/202623_2.png) [@aplund](https://discourse.julialang.org/u/aplund)\
**Post date:** [October 26, 2023, 5:58am UTC](https://discourse.julialang.org/t/how-to-get-pde-solutions-when-doing-foward-time-steps/105419/1 "2023-10-26T05:58:58Z")

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I’ve been trying to get the hang of solving DEs in Julia (i.e. new user). As a simple task, I tried to set up a 1-d Schrödinger equation and perform a time evolution. Here’s my effort so far:

```julia
using ModelingToolkit
using MethodOfLines
using DomainSets

@variables t [bounds = (0.0, Inf)]
@variables x [bounds = (-5.0, 5.0)]
@variables ur(..), ui(..)

Dx = Differential(x)
Dt = Differential(t)

equationri = [
    Dt(ur(t,x)) ~ -(1/2)*(Dx^2)(ui(t,x)), # real LHS
    Dt(ui(t,x)) ~ (1/2)*(Dx^2)(ur(t,x)), # imag LHS
]

domains=[
    t ∈ (0.0, Inf),
    x ∈ (-5.0,5.0)
]

boundary_conditions=[
    ur(0.0,x) ~ cos(x) * exp(-x^2/4) / (2π)^(1/4),
    ui(0.0,x) ~ sin(x) * exp(-x^2/4) / (2π)^(1/4),
    ur(t,-5.0) ~ 0.0,
    ui(t,-5.0) ~ 0.0,
    ur(t,5.0) ~ 0.0,
    ui(t,5.0) ~ 0.0,
]

@named schro_system = PDESystem(
    equationri,
    boundary_conditions,
    domains,
    [t, x],
    [ur(t,x), ui(t,x)]
)

dx=0.1
discretization = MOLFiniteDifference([x => dx], t; approx_order = 2)
problem = discretize(schro_system, discretization)

using OrdinaryDiffEq

st=init(problem, Tsit5()) # here we are initialized
# step with `step!(st, 0.05, true)` (don't know why kwarg doesn't work)
# Solution in st.sol

```

The problem I’m having is that I can not get the solution out of the `st.sol` ODESolution object. It seems to suggest that it is in the index `u`, but “u” is undefined. Doing an index of `ur(t,x)` just gives

```julia
ERROR: ArgumentError: ur(t, x) is neither an observed nor a state variable.

```

which is a somewhat confusing message.

The documentation here seems somewhat lacking. There is some warning about the “old” system for discretising. But the “new” system doesn’t actually seem to work.

So, I guess my questions are: is the above code snippet correct usage? If so, how can I get the solution read out after perfoming forwards integration step?

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<div class="post-metadata">

**Author:** ![xtalax](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/xtalax/32/35293_2.png) [@xtalax](https://discourse.julialang.org/u/xtalax)\
**Post date:** [October 28, 2023, 4:21pm UTC](https://discourse.julialang.org/t/how-to-get-pde-solutions-when-doing-foward-time-steps/105419/2 "2023-10-28T16:21:51Z")

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Yes, the initialising the solution it isn’t yet wrapped, you have to use [the old method](https://docs.sciml.ai/MethodOfLines/stable/get_grid/). Are you sure you don’t want to just `solve` the problem? Init is called internally in the solve method so you don’t need to do it manually, what are you trying to do?

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<div class="post-metadata">

**Author:** ![aplund](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/aplund/32/202623_2.png) [@aplund](https://discourse.julialang.org/u/aplund)\
**Post date:** [October 28, 2023, 11:06pm UTC](https://discourse.julialang.org/t/how-to-get-pde-solutions-when-doing-foward-time-steps/105419/3 "2023-10-28T23:06:38Z")

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I have been trying to do an animated plot with Makie of the solution in 1-d. I’ve been fiddling with this and eventually found using the “old method” works. I was initially trying to only integrate up to the time step in which a single frame would be drawn.