# DifferentialEquations.jl: split differential equation with explicit propagators
**URL:** https://discourse.julialang.org/t/differentialequations-jl-split-differential-equation-with-explicit-propagators/7572
**Category:** Numerics
**Tags:** diffeq
**Created:** [December 6, 2017, 5:01pm UTC](https://discourse.julialang.org/t/differentialequations-jl-split-differential-equation-with-explicit-propagators/7572 "2017-12-06T17:01:48Z")
**Posts on this page:** 4
**Page:** 1
### Author:  [@antoine-levitt](https://discourse.julialang.org/u/antoine-levitt)
#### Post date: [December 6, 2017, 5:01pm UTC](https://discourse.julialang.org/t/differentialequations-jl-split-differential-equation-with-explicit-propagators/7572/1 "2017-12-06T17:01:48Z")
I have a PDE that is (not exactly but close to) du/dt + v \cdot du/dx = A(x)u, where x is 2D, u is a vector with two components and A is a 2x2 matrix. The initial data possibly has shocks (discontinuities). Right now I use finite differences + DifferentialEquations, but the numerical diffusion is killing me. I thought about using those fancy hyperbolic schemes, but they look complicated. Alternatively, I can use a splitting method (I know how to integrate without v, and I know how to integrate without A). Can I fit this into DifferentialEquations.jl to take care of the higher-order/timestepping for me? I did not find it in the doc, I just ask in case it’s hidden somewhere.
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [December 6, 2017, 5:14pm UTC](https://discourse.julialang.org/t/differentialequations-jl-split-differential-equation-with-explicit-propagators/7572/2 "2017-12-06T17:14:09Z")
This is something we’re still working on. Operator splitting methods are in a very rudimentary state:
[http://docs.juliadiffeq.org/latest/solvers/split\_ode\_solve.html](http://docs.juliadiffeq.org/latest/solvers/split_ode_solve.html)
It’s one of the next big sets of algorithms we are working on though. Until we have those kinds of methods, I would recommend just using the SSP methods
[http://docs.juliadiffeq.org/latest/solvers/ode\_solve.html#Explicit-Strong-Stability-Preserving-Runge-Kutta-Methods-for-Hyperbolic-PDEs-(Conservation-Laws)-1](http://docs.juliadiffeq.org/latest/solvers/ode_solve.html#Explicit-Strong-Stability-Preserving-Runge-Kutta-Methods-for-Hyperbolic-PDEs-(Conservation-Laws)-1)
with a sufficient limiter. The other types of methods which would do well are the IMEX schemes which we don’t have yet:
[https://github.com/JuliaDiffEq/OrdinaryDiffEq.jl/issues/191](https://github.com/JuliaDiffEq/OrdinaryDiffEq.jl/issues/191)
So yeah, probably not satisfactory but it’s what we got. You can also just try throwing it into Sundials’ `CVODE_BDF` which might do well enough.
---
### Author:  [@ChrisRackauckas](https://discourse.julialang.org/u/ChrisRackauckas)
#### Post date: [December 6, 2017, 6:29pm UTC](https://discourse.julialang.org/t/differentialequations-jl-split-differential-equation-with-explicit-propagators/7572/3 "2017-12-06T18:29:53Z")
Another thing that helps with numerical differentiation is to use a higher order upwinding scheme. You can check out DiffEqOperators.jl for some of the operators for that:
> **[GitHub - SciML/DiffEqOperators.jl: Linear operators for discretizations of...](https://github.com/SciML/DiffEqOperators.jl)**
>
> Linear operators for discretizations of differential equations and scientific machine learning (SciML) - GitHub - SciML/DiffEqOperators.jl: Linear operators for discretizations of differential equa...
The documentation there is still lacking though. But if you really need to get rid of numerical diffusion, you may need to use a better spatial discretization.
---
### Author:  [@antoine-levitt](https://discourse.julialang.org/u/antoine-levitt)
#### Post date: [December 6, 2017, 7:45pm UTC](https://discourse.julialang.org/t/differentialequations-jl-split-differential-equation-with-explicit-propagators/7572/4 "2017-12-06T19:45:03Z")
Actually I thought I knew how to solve without the A on a grid, but of course that’s bullshit: if `v*delta_t` is not a multiple of `delta_x` there’s no obvious way to transport the solution. So as you say I probably need to rethink the discretization. Still since in my case v is constant, maybe I can impose that `v*delta_t = delta_x` and solve exactly. I wonder if that works (and if I’m rediscovering Lattice Boltzmann or something like that). In any case I definitely need to read some literature. Thanks for the input!