New algorithms can either be added by extending one of the current solver (or add-on packages), or by contributing a new package to the organization. If it's a new problem (a new PDE, a new type of differential equation, a new subclass of problems for which special methods exist, etc.) then the problem and solution types should be added to DiffEqBase first.

After the problem and solutions are defined, the __solve method should be implemented. It should take in keyword arguments which match the common interface (implement "as many as possible"). One should note and document the amount of compatibility with the common interface and Julia-defined types. After that, testing should be done using DiffEqDevTools. Convergence tests and benchmarks should be included to show the effectiveness of the algorithm and the correctness. Do not worry if the algorithm is not "effective": the implementation can improve over time and some algorithms useful just for the comparison they give!

After some development, one may want to document the algorithm in DiffEqBenchmarks and DiffEqTutorials.

## Adding new algorithms to OrdinaryDiffEq

This recipe has been used to add the strong stability preserving Runge-Kutta methods SSPRK22, SSPRK33, and SSPRK104 to OrdinaryDiffEq. SSPRK22 will be used as an example.

• To create a new solver, two (three) types have to be created. The first is the algorithm SSPRK22 used for dispatch, the other ones are the corresponding caches SSPRK22Cache (for inplace updates) and SSPRK22ConstantCache.
• The algorithm is defined in algorithms.jl as struct SSPRK22 <: OrdinaryDiffEqAlgorithm end. Although it does not have the FSAL property, this is set to true since the derivative at the start and the end of the interval are used for the Hermite interpolation, and so this is FSAL'd so that way only a single extra function evaluation occurs over the whole integration. This is done in alg_utils.jl via isfsal(alg::SSPRK22) = true. Additionally, the order is set in the same file via alg_order(alg::SSPRK22) = 2.
• The algorithm SSPRK22 is exported in OrdinaryDiffEq.jl.
• In caches.jl, the two cache types SSPRK22Cache (for inplace updates) and SSPRK22ConstantCache are defined, similarly to the other ones. Note: u_cache(c::SSPRK22Cache) = () and du_cache(c::SSPRK22Cache) = (c.k,c.du,c.fsalfirst) return the parts of the modifiable cache that are changed if the size of the ODE changes.
• A new file perform_step/ssprk_perform_step.jl has been used for the new implementations. For both types of caches, the functions initialize! and perform_step! are defined there.
• Finally, tests are added. A new file test/ode/ode_ssprk_tests.jl is created and included in tests/runtests.jl via @time @testset "SSPRK Tests" begin include("ode/ode_ssprk_tests.jl") end.
• Additionally, regression tests for the dense output are added in test/ode/ode_dense_tests.jl.

For more details, refer to https://github.com/JuliaDiffEq/OrdinaryDiffEq.jl/pull/40

### Self-Contained Example

using OrdinaryDiffEq
import OrdinaryDiffEq:
OrdinaryDiffEqAlgorithm, OrdinaryDiffEqMutableCache, OrdinaryDiffEqConstantCache,
alg_order, alg_cache, initialize!, perform_step!, trivial_limiter!, constvalue,

struct RK_ALG{StageLimiter,StepLimiter} <: OrdinaryDiffEq.OrdinaryDiffEqAlgorithm
stage_limiter!::StageLimiter
step_limiter!::StepLimiter
end
RK_ALG(stage_limiter! = trivial_limiter!) = RK_ALG(stage_limiter!, trivial_limiter!)
export RK_ALG
alg_order(alg::RK_ALG) = 3

@cache struct RK_ALGCache{uType,rateType,StageLimiter,StepLimiter,TabType} <: OrdinaryDiffEqMutableCache
u::uType
uprev::uType
k::rateType
tmp::uType
u₂::uType
fsalfirst::rateType
stage_limiter!::StageLimiter
step_limiter!::StepLimiter
tab::TabType
end

struct RK_ALGConstantCache{T,T2} <: OrdinaryDiffEqConstantCache
α40::T
α41::T
α43::T
α62::T
α65::T
β10::T
β21::T
β32::T
β43::T
β54::T
β65::T
c1::T2
c2::T2
c3::T2
c4::T2
c5::T2
end

function RK_ALGConstantCache(T, T2)
α40 = T(0.476769811285196)
α41 = T(0.098511733286064)
α43 = T(0.424718455428740)
α62 = T(0.155221702560091)
α65 = T(0.844778297439909)
β10 = T(0.284220721334261)
β21 = T(0.284220721334261)
β32 = T(0.284220721334261)
β43 = T(0.120713785765930)
β54 = T(0.284220721334261)
β65 = T(0.240103497065900)
c1 = T2(0.284220721334261)
c2 = T2(0.568441442668522)
c3 = T2(0.852662164002783)
c4 = T2(0.510854218958172)
c5 = T2(0.795074940292433)

RK_ALGConstantCache(α40, α41, α43, α62, α65, β10, β21, β32, β43, β54, β65, c1, c2, c3, c4, c5)
end

function alg_cache(alg::RK_ALG,u,rate_prototype,uEltypeNoUnits,uBottomEltypeNoUnits,tTypeNoUnits,uprev,uprev2,f,t,dt,reltol,p,calck,::Val{true})
tmp = similar(u)
u₂ = similar(u)
k = zero(rate_prototype)
fsalfirst = zero(rate_prototype)
tab = RK_ALGConstantCache(real(uBottomEltypeNoUnits), real(tTypeNoUnits))
RK_ALGCache(u,uprev,k,tmp,u₂,fsalfirst,alg.stage_limiter!,alg.step_limiter!,tab)
end

function alg_cache(alg::RK_ALG,u,rate_prototype,uEltypeNoUnits,uBottomEltypeNoUnits,tTypeNoUnits,uprev,uprev2,f,t,dt,reltol,p,calck,::Val{false})
RK_ALGConstantCache(real(uBottomEltypeNoUnits), real(tTypeNoUnits))
end

function initialize!(integrator,cache::RK_ALGConstantCache)
integrator.fsalfirst = integrator.f(integrator.uprev,integrator.p,integrator.t) # Pre-start fsal
integrator.destats.nf += 1
integrator.kshortsize = 1
integrator.k = typeof(integrator.k)(undef, integrator.kshortsize)

# Avoid undefined entries if k is an array of arrays
integrator.fsallast = zero(integrator.fsalfirst)
integrator.k[1] = integrator.fsalfirst
end

@unpack t,dt,uprev,u,f,p = integrator
@unpack α40,α41,α43,α62,α65,β10,β21,β32,β43,β54,β65,c1,c2,c3,c4,c5 = cache

# u1 -> stored as u
u = uprev + β10 * dt * integrator.fsalfirst
k = f(u, p, t+c1*dt)
# u2
u₂ = u + β21 * dt * k
k = f(u₂,p,t+c2*dt)
# u3
tmp = u₂ + β32 * dt * k
k = f(tmp, p, t+c3*dt)
# u4
tmp = α40 * uprev + α41 * u + α43 * tmp + β43 * dt * k
k = f(tmp, p, t+c4*dt)
# u5
tmp = tmp + β54 * dt * k
k = f(tmp, p, t+c5*dt)
# u
u = α62 * u₂ + α65 * tmp + β65 * dt * k

integrator.fsallast = f(u, p, t+dt) # For interpolation, then FSAL'd
integrator.destats.nf += 6
integrator.k[1] = integrator.fsalfirst
integrator.u = u
end

function initialize!(integrator,cache::RK_ALGCache)
@unpack k,fsalfirst = cache
integrator.fsalfirst = fsalfirst
integrator.fsallast = k
integrator.kshortsize = 1
resize!(integrator.k, integrator.kshortsize)
integrator.k[1] = integrator.fsalfirst
integrator.f(integrator.fsalfirst,integrator.uprev,integrator.p,integrator.t) # FSAL for interpolation
integrator.destats.nf += 1
end

@unpack t,dt,uprev,u,f,p = integrator
@unpack k,tmp,u₂,fsalfirst,stage_limiter!,step_limiter! = cache
@unpack α40,α41,α43,α62,α65,β10,β21,β32,β43,β54,β65,c1,c2,c3,c4,c5 = cache.tab

# u1 -> stored as u
@.. u = uprev + β10 * dt * integrator.fsalfirst
stage_limiter!(u, f, p, t+c1*dt)
f( k,  u, p, t+c1*dt)
# u2
@.. u₂ = u + β21 * dt * k
stage_limiter!(u₂, f, p, t+c2*dt)
f(k,u₂,p,t+c2*dt)
# u3
@.. tmp = u₂ + β32 * dt * k
stage_limiter!(tmp, f, p, t+c3*dt)
f( k,  tmp, p, t+c3*dt)
# u4
@.. tmp = α40 * uprev + α41 * u + α43 * tmp + β43 * dt * k
stage_limiter!(tmp, f, p, t+c4*dt)
f( k,  tmp, p, t+c4*dt)
# u5
@.. tmp = tmp + β54 * dt * k
stage_limiter!(tmp, f, p, t+c5*dt)
f( k,  tmp, p, t+c5*dt)
# u
@.. u = α62 * u₂ + α65 * tmp + β65 * dt * k
stage_limiter!(u, f, p, t+dt)
step_limiter!(u, f, p, t+dt)
integrator.destats.nf += 6
f( k,  u, p, t+dt)
end

#oop test
f = ODEFunction((u,p,t)->1.01u,
analytic = (u0,p,t) -> u0*exp(1.01t))
prob = ODEProblem(f,1.01,(0.0,1.0))
sol = solve(prob,RK_ALG(),dt=0.1)

using Plots
plot(sol)
plot(sol,denseplot=false,plot_analytic=true)

using DiffEqDevTools
dts = (1/2) .^ (8:-1:1)
sim = test_convergence(dts,prob,RK_ALG())
sim.𝒪est[:final]
plot(sim)

# Example of a good one!
sim = test_convergence(dts,prob,BS3())
sim.𝒪est[:final]
plot(sim)

#iip test
f = ODEFunction((du,u,p,t)->(du .= 1.01.*u),
analytic = (u0,p,t) -> u0*exp(1.01t))
prob = ODEProblem(f,[1.01],(0.0,1.0))
sol = solve(prob,RK_ALG(),dt=0.1)

plot(sol)
plot(sol,denseplot=false,plot_analytic=true)

dts = (1/2) .^ (8:-1:1)
sim = test_convergence(dts,prob,RK_ALG())
sim.𝒪est[:final]
plot(sim)

# Example of a good one!
sim = test_convergence(dts,prob,BS3())
sim.𝒪est[:final]
plot(sim)

The exponential algorithms follow the same recipe as the general algorithms, but there are automation utilities that make this easier. It is recommended that you refer to one of the model algorithms for reference:

• For traditional exponential Runge-Kutta type methods (that come with a corresponding Butcher table), refer to ETDRK2.
• For adaptive exponential Rosenbrock type methods, refer to Exprb32.
• For exponential propagation iterative Runge-Kutta methods (EPIRK), refer to EPIRK5P1.

The first two classes support two modes of operation: operator caching and Krylov approximation. The perform_step! method in perform_step/exponential_rk_perform_step.jl, as a result, is split into two branches depending on whether alg.krylov is true. The caching branch utilizes precomputed operators, which are calculated by the expRK_operators method in caches/linear_nonlinear_caches.jl. Both expRK_operators and the arnoldi/phiv methods in perform_step! comes from the ExponentialUtilities package.

The EPIRK methods can only use Krylov approximation, and unlike the previous two they use the timestepping variant phiv_timestep. The timestepping method follows the convention of Neisen & Wright, and can be toggled to use adaptation by alg.adaptive_krylov.

Although the exponential integrators (especially the in-place version) can seem complex, they share similar structures. The infrastructure for the existing exponential methods utilize the fact to reduce boilerplate code. In particular, the cache construction code in caches/linear_nonlinear_caches.jl and the initialize! method in perform_step/exponential_rk_perform_step.jl can be mostly automated and only perform_step! needs implementing.

Finally, to construct tests for the new exponential algorithm, append the new algorithm to the corresponding algorithm class in test/linear_nonlinear_convergence_tests.jl and test/linear_nonlinear_krylov_tests.jl.