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tutorials/advanced/02-advanced_ODE_solving.jmd

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Choosing a good solver is required for getting top notch speed. General
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recommendations can be found on the solver page (for example, the
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[ODE Solver Recommendations](http://docs.juliadiffeq.org/latest/solvers/ode_solve.html)).
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[ODE Solver Recommendations](http://docs.juliadiffeq.org/dev/solvers/ode_solve.html)).
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The current recommendations can be simplified to a Rosenbrock method
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(`Rosenbrock23` or `Rodas5`) for smaller (<50 ODEs) problems, ESDIRK methods
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for slightly larger (`TRBDF2` or `KenCarp4` for <2000 ODEs), and Sundials
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### Check Out the Speed FAQ
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See [this FAQ](http://docs.juliadiffeq.org/latest/basics/faq.html#Performance-1)
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See [this FAQ](http://docs.juliadiffeq.org/dev/basics/faq.html#Performance-1)
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for information on common pitfalls and how to improve performance.
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### Setting Up Your Julia Installation for Speed
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When possible, use GPUs. If your ODE system is small and you need to solve it
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with very many different parameters, see the
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[ensembles interface](http://docs.juliadiffeq.org/latest/features/ensemble.html)
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[ensembles interface](http://docs.juliadiffeq.org/dev/features/ensemble.html)
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and [DiffEqGPU.jl](https://github.com/JuliaDiffEq/DiffEqGPU.jl). If your problem
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is large, consider using a [CuArray](https://github.com/JuliaGPU/CuArrays.jl)
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for the state to allow for GPU-parallelism of the internal linear algebra.
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```
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For more information on linear solver choices, see the
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[linear solver documentation](http://docs.juliadiffeq.org/latest/features/linear_nonlinear.html).
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[linear solver documentation](http://docs.juliadiffeq.org/dev/features/linear_nonlinear.html).
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On this problem, handling the sparsity correctly seemed to give much more of a
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speedup than going to a Krylov approach, but that can be dependent on the problem
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### Adding a Preconditioner
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The [linear solver documentation](http://docs.juliadiffeq.org/latest/features/linear_nonlinear.html#IterativeSolvers.jl-Based-Methods-1)
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The [linear solver documentation](http://docs.juliadiffeq.org/dev/features/linear_nonlinear.html#IterativeSolvers.jl-Based-Methods-1)
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shows how you can add a preconditioner to the GMRES. For example, you can
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use packages like [AlgebraicMultigrid.jl](https://github.com/JuliaLinearAlgebra/AlgebraicMultigrid.jl)
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to add an algebraic multigrid (AMG) or [IncompleteLU.jl](https://github.com/haampie/IncompleteLU.jl)
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While much of the setup makes the transition to using Sundials automatic, there
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are some differences between the pure Julia implementations and the Sundials
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implementations which must be taken note of. These are all detailed in the
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[Sundials solver documentation](http://docs.juliadiffeq.org/latest/solvers/ode_solve.html#Sundials.jl-1),
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[Sundials solver documentation](http://docs.juliadiffeq.org/dev/solvers/ode_solve.html#Sundials.jl-1),
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but here we will highlight the main details which one should make note of.
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Defining a sparse matrix and a Jacobian for Sundials works just like any other
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```
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Details for setting up a preconditioner with Sundials can be found at the
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[Sundials solver page](http://docs.juliadiffeq.org/latest/solvers/ode_solve.html#Sundials.jl-1).
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[Sundials solver page](http://docs.juliadiffeq.org/dev/solvers/ode_solve.html#Sundials.jl-1).
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## Handling Mass Matrices
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Note that if your mass matrix is singular, i.e. your system is a DAE, then you
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need to make sure you choose
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[a solver that is compatible with DAEs](http://docs.juliadiffeq.org/latest/solvers/dae_solve.html#Full-List-of-Methods-1)
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[a solver that is compatible with DAEs](http://docs.juliadiffeq.org/dev/solvers/dae_solve.html#Full-List-of-Methods-1)

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