# Test Problems

For every problem, one can turn it into a test problem by adding analytical to the DEFunction.

## No Analytical Solution

However, in many cases the analytical solution cannot be found, and therefore one uses a low-tolerance calculation as a stand-in for a solution. The JuliaDiffEq ecosystem supports this through the TestSolution type in DiffEqDevTools. There are three constructors. The code is simple, so here it is:

type TestSolution <: DESolution
t
u
interp
dense
end
(T::TestSolution)(t) = T.interp(t)
TestSolution(t,u) = TestSolution(t,u,nothing,false)
TestSolution(t,u,interp) = TestSolution(t,u,interp,true)
TestSolution(interp::DESolution) = TestSolution(nothing,nothing,interp,true)

This acts like a solution. When used in conjunction with apprxtrue:

appxtrue(sol::AbstractODESolution,sol2::TestSolution)

you can use it to build a TestSolution from a problem (like ODETestSolution) which holds the errors If you only give it t and u, then it can only calculate the final error. If the TestSolution has an interpolation, it will define timeseries and dense errors.

(Note: I would like it so that way the timeseries error will be calculated on the times of sol.t in sol2.t which would act nicely with tstops and when interpolations don't exist, but haven't gotten to it!)

These can then be passed to other functionality. For example, the benchmarking functions allow one to set appxsol which is a TestSolution for the benchmark solution to calculate errors against, and error_estimate allows one to choose which error estimate to use in the benchmarking (defaults to :final).

DiffEqDevTools.appxtrueFunction

appxtrue(sol::AbstractODESolution,sol2::TestSolution)

Uses the interpolant from the higher order solution sol2 to approximate errors for sol. If sol2 has no interpolant, only the final error is calculated.

appxtrue(sol::AbstractODESolution,sol2::AbstractODESolution)

Uses the interpolant from the higher order solution sol2 to approximate errors for sol. If sol2 has no interpolant, only the final error is calculated.