# Solving a complex second order PDEs coupling with ODEs

**URL:** <https://discourse.julialang.org/t/solving-a-complex-second-order-pdes-coupling-with-odes/128792>\
**Category:** General Usage\
**Tags:** question, diffeq\
**Created:** [May 7, 2025, 12:51pm UTC](https://discourse.julialang.org/t/solving-a-complex-second-order-pdes-coupling-with-odes/128792 "2025-05-07T12:51:56Z")\
**Posts on this page:** 4\
**Page:** 2

<div class="post-metadata">

**Author:** ![k\_zhou](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/k_zhou/32/216695_2.png) [@k\_zhou](https://discourse.julialang.org/u/k_zhou)\
**Post date:** [May 22, 2025, 9:18am UTC](https://discourse.julialang.org/t/solving-a-complex-second-order-pdes-coupling-with-odes/128792/21 "2025-05-22T09:18:45Z")

</div>

Here I just discretize the dE/dz item by du[i, 9] = (u[i, 7] - u[i-1, 7]) / dx and du[i, 10] = -(u[i+1, 8] - u[i, 8]) / dx. There are no spatial terms elsewhere in the equations (u[1] to u[6]).

---

<div class="post-metadata">

**Author:** ![ChrisRackauckas](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/chrisrackauckas/32/77_2.png) [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)\
**Post date:** [May 24, 2025, 10:19pm UTC](https://discourse.julialang.org/t/solving-a-complex-second-order-pdes-coupling-with-odes/128792/22 "2025-05-24T22:19:09Z")

</div>

Are you sure you got the winding direction right? [Upwind scheme - Wikipedia](https://en.wikipedia.org/wiki/Upwind_scheme)

---

<div class="post-metadata">

**Author:** ![k\_zhou](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/k_zhou/32/216695_2.png) [@k\_zhou](https://discourse.julialang.org/u/k_zhou)\
**Post date:** [May 28, 2025, 2:51am UTC](https://discourse.julialang.org/t/solving-a-complex-second-order-pdes-coupling-with-odes/128792/23 "2025-05-28T02:51:26Z")

</div>

I check the wiki, It looks like right 😅

```julia
du[i, 9] = (u[i, 7] - u[i-1, 7]) / dx # dE+/dz
du[i, 10] = -(u[i+1, 8] - u[i, 8]) / dx #-dE-/dz

```

---

<div class="post-metadata">

**Author:** ![k\_zhou](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/k_zhou/32/216695_2.png) [@k\_zhou](https://discourse.julialang.org/u/k_zhou)\
**Post date:** [June 16, 2025, 8:01am UTC](https://discourse.julialang.org/t/solving-a-complex-second-order-pdes-coupling-with-odes/128792/24 "2025-06-16T08:01:45Z")

</div>

Updating!  
I have found a method to convert the dispersion item to the second order spatial difference item. By using central difference, the problem can be solved now（Tsit5 is faster than the ROCK4 by ~20% in my problem).

Thx for your help 😄

[Previous page](https://discourse.julialang.org/t/solving-a-complex-second-order-pdes-coupling-with-odes/128792.md?page=1)