# DifferentialEquations.jl v6.8.0: Advanced Stiff Differential Equation Solving

This release covers the completion of another successful summer. We have now completed a new round of tooling for solving large stiff and sparse differential equations. Most of this is covered in the exciting….

## New Tutorial: Solving Stiff Equations for Advanced Users!

That is right, we now have a new tutorial added to the documentation on solving stiff differential equations. This tutorial goes into depth, showing how to use our recent developments to do things like automatically detect and optimize a solver with respect to sparsity pattern, or automatically symbolically calculate a Jacobian from a numerical code. This should serve as a great resource for the advanced users who want to know how to get started with those finer details like sparsity patterns and mass matrices.

## Automatic Colorization and Optimization for Structured Matrices

As showcased in the tutorial, if you have `jac_prototype`

be a structured matrix,
then the `colorvec`

is automatically computed, meaning that things like
`BandedMatrix`

are now automatically optimized. The default linear solvers make
use of their special methods, meaning that DiffEq has full support for these
structured matrix objects in an optimal manner.

## Implicit Extrapolation and Parallel DIRK for Stiff ODEs

At the tail end of the summer, a set of implicit extrapolation methods were completed. We plan to parallelize these over the next year, seeing what can happen on small stiff ODEs if parallel W-factorizations are allowed.

## Automatic Conversion of Numerical to Symbolic Code with Modelingtoolkitize

This is just really cool and showcased in the new tutorial. If you give us a function for numerically computing the ODE, we can now automatically convert said function into a symbolic form in order to compute quantities like the Jacobia and then build a Julia code for the generated Jacobian. Check out the new tutorial if you’re curious, because although it sounds crazy… this is now a standard feature!

## GPU-Optimized Sparse (Colored) Automatic and Finite Differentiation

SparseDiffTools.jl and DiffEqDiffTools.jl were made GPU-optimized, meaning that the stiff ODE solvers now do not have a rate-limiting step at the Jacobian construction.

## DiffEqBiological.jl: Homotopy Continuation

DiffEqBiological got support for automatic bifurcation plot generation by connecting with HomotopyContinuation.jl. See the new tutorial

## Greatly improved delay differential equation solving

David Widmann (@devmotion) greatly improved the delay differential equation solver’s implicit step handling, along with adding a bunch of tests to show that it passes the special RADAR5 test suite!

## Color Differentiation Integration with Native Julia DE Solvers

The `ODEFunction`

, `DDEFunction`

, `SDEFunction`

, `DAEFunction`

, etc. constructors
now allow you to specify a color vector. This will reduce the number of `f`

calls required to compute a sparse Jacobian, giving a massive speedup to the
computation of a Jacobian and thus of an implicit differential equation solve.
The color vectors can be computed automatically using the SparseDiffTools.jl
library’s `matrix_colors`

function. Thank JSoC student Langwen Huang
(@huanglangwen) for this contribution.

## Improved compile times

Compile times should be majorly improved now thanks to work from David Widmann (@devmotion) and others.

# Next Directions

Our current development is very much driven by the ongoing GSoC/JSoC projects, which is a good thing because they are outputting some really amazing results!

Here’s some things to look forward to:

- Automated matrix-free finite difference PDE operators
- Jacobian reuse efficiency in Rosenbrock-W methods
- Native Julia fully implicit ODE (DAE) solving in OrdinaryDiffEq.jl
- High Strong Order Methods for Non-Commutative Noise SDEs
- Stochastic delay differential equations