# DifferentialEquations.jl 4.6: Global Sensitivity Analysis, Variable Order Adams

Tons of improvements due to Google Summer of Code. Here’s what’s happened.

## Global Sensitivty Analysis (Morris, Sobol)

GSoC student Vaibhav Dixit (@Vaibhavdixit02) added global sensitivity analysis (GSA) methods to DiffEqSensitivity.jl. GSA quantifies the effects of the parameters on the solution of the ODE. The implementation of these methods are on any generic f(p), so this may be refactored into an external non-DiffEq library. The Morris method and the Sobol method, two of the most commonly used GSA methods, are part of this implementation. Other methods, such as PRC, SRC, FAST, eFAST, etc. are coming soon.

## Variable time step variable order Adams methods (VCABM)

GSoC student Shubham Maddhashiya (@sipah00) added a variable-coefficient form implementation of the variable order variable time step Adams-Bashforth-Moulton method. This implementation matches the classic DDEABM software of Shampine which specializes in its ability to utilize the higher orders for larger time steps and high efficiency on less stiff equations. Our benchmarks show efficiency improvements over not only DDEABM, but also over the Sundials’ CVODE Adams implementation (not surprised here though: Hairer had showed before that a variable coefficient form can give more effective order selection) and the Runge-Kutta methods on some problems. This makes it a good native Julia replacement to those classic methods, not only for large non-stiff ODE systems, but also on many other smooth non-stiff systems.

For example, here is the new method applied to the Pleiades Problem:

In the previous notebook, the fastest methods were the Vern6, Vern7, and Sundials CVODE methods. Now this method seems to be a new contender in this field.

## High Stiff Order Exponential Runge-Kutta and EPIRK Methods

GSoC student Xingjian Guo (@MSeeker1340) added the HochOst4 and Exp4 algorithms which is an exponential Runge-Kutta algorithm with stiff order 4. Unlike other high order ExpRK methods which have issues with order loss when solving a problem with high stiffness, this method retains its order even on this difficult class of problems. Note that the classic 4th order exponential integrator, ETDRK4, is only stiff order 1 making it perform less effectively than HochOst4 and Exp4 on the highly stiff PDE discretizations that it was designed for. Xingjian is continuing this line of development with adaptive ExpRK methods and more high stiff order EPIRK methods.

## Low Order IMEX Methods

GSoC student Shubham Maddhashiya (@sipah00) added some low order IMEX methods to the OrdinaryDiffEq.jl solver suite. These methods are common methods for solving PDEs, especially spectral discretizations of PDEs. Crank-Nicholson Adams-Bashforth 2 (CNAB2), Crank-Nicholson Leapfrog (CNLF), and an Implicit-Explicit Euler method (IMEXEuler) are all available on the common interface. In many cases one may want to utilize the higher order methods, but there are still many uses for these. For example, implicit Euler is the only method with an infinite strong stability preserving (SSP) coefficient, meaning that it can be much more stable than other methods for hyperbolic PDEs. This IMEXEuler can be an easy way to utilize a more efficient than the standard IMEXEuler.

# In development

A lot of the next developments will come from our GSoC students. Here’s a list of things we are aiming for:

• Quasi-constant stepsize variable coefficient BDF and NDF and IMEX BDF (SBDF) integrators. Both fixed and variable order.

• High order EPIRK adaptive exponential Runge-Kutta methods.

• Fixed Leading Coefficient (FLC) form Nordsieck BDF integrators.

• SABDF2, which is a strong order 0.5 adaptive BDF2 implementation for stochastic differential equations which is order 2 for small noise SDEs. This will be the first implicit adaptive integrator for small noise SDEs and will be a great choice for SPDEs.

• Yiannis Simillides (@ysimillides) keeps making improvements to FEniCS.jl. At this part a large portion (a majority?) of the tutorial works from Julia. Integration with native Julia tools like Makie.jl is in progress.

• Mikhail Vaganov (@Mikhail-Vaganov) is making good progress on his N-body modeling language. This will make it easy to utilize DiffEq as a backend for molecular dynamics simulation. Follow the progress in DiffEqPhysics.jl

And here’s a quick view of the rest of our “in development” list:

• Preconditioner choices for Sundials methods
• Adaptivity in the MIRK BVP solvers
• More general Banded and sparse Jacobian support outside of Sundials
• Function input for initial conditions and time span (u0(p,t0))
• LSODA integrator interface

# Projects

Are you a student who is interested in working on differential equations software and modeling? If so, please get in touch with us since we may have some funding after August for some student developers to contribute towards some related goals. It’s not guaranteed yet, but getting in touch never hurts!