# Solution Handling

## Accessing the Values

The solution type has a lot of built in functionality to help analysis. For example, it has an array interface for accessing the values. Internally, the solution type has two important fields:

1. u which holds the Vector of values at each timestep
2. t which holds the times of each timestep.

Different solution types may add extra information as necessary, such as the derivative at each timestep du or the spatial discretization x, y, etc.

## Array Interface

Instead of working on the Vector{uType} directly, we can use the provided array interface.

sol[j]

to access the value at timestep j (if the timeseries was saved), and

sol.t[j]

to access the value of t at timestep j. For multi-dimensional systems, this will address first by component and lastly by time, and thus

sol[i,j]

will be the ith component at timestep j. Hence, sol[j][i] == sol[i, j]. This is done because Julia is column-major, so the leading dimension should be contiguous in memory. If the independent variables had shape (for example, was a matrix), then i is the linear index. We can also access solutions with shape:

sol[i,k,j]

gives the [i,k] component of the system at timestep j. The colon operator is supported, meaning that

sol[i,:]

gives the timeseries for the ith component.

## Using the AbstractArray Interface

The AbstractArray interface can be directly used. For example, for a vector system of variables sol[i,j] is a matrix with rows being the variables and columns being the timepoints. Operations like sol' will transpose the solution type. Functionality written for AbstractArrays can directly use this. For example, the Base cov function computes correlations amongst columns, and thus:

cov(sol)

computes the correlation of the system state in time, whereas

cov(sol,2)

computes the correlation between the variables. Similarly, mean(sol,2) is the mean of the variable in time, and var(sol,2) is the variance. Other statistical functions and packages which work on AbstractArray types will work on the solution type.

At anytime, a true Array can be created using Array(sol).

## Interpolations and Calculating Derivatives

If the solver allows for dense output and dense=true was set for the solving (which is the default), then we can access the approximate value at a time t using the command

sol(t)

Note that the interpolating function allows for t to be a vector and uses this to speed up the interpolation calculations. The full API for the interpolations is

sol(t,deriv=Val{0};idxs=nothing,continuity=:left)

The optional argument deriv lets you choose the number n derivative to solve the interpolation for, defaulting with n=0. Note that most of the derivatives have not yet been implemented (though it's not hard, it just has to be done by hand for each algorithm. Open an issue if there's a specific one you need). continuity describes whether to satisfy left or right continuity when a discontinuity is saved. The default is :left, i.e. grab the value before the callback's change, but can be changed to :right. idxs allows you to choose the indices the interpolation should solve for. For example,

sol(t,idxs=1:2:5)

will return a Vector of length 3 which is the interpolated values at t for components 1, 3, and 5. idxs=nothing, the default, means it will return every component. In addition, we can do

sol(t,idxs=1)

and it will return a Number for the interpolation of the single value. Note that this interpolation only computes the values which are requested, and thus it's much faster on large systems to use this rather than computing the full interpolation and using only a few values.

In addition, there is an inplace form:

sol(out,t,deriv=Val{0};idxs=nothing,continuity=:left)

which will write the output to out. This allows one to use pre-allocated vectors for the output to improve the speed even more.

## Comprehensions

The solver interface also gives tools for using comprehensions over the solution. Using the tuples(sol) function, we can get a tuple for the output at each timestep. This allows one to do the following:

[t+2u for (u,t) in tuples(sol)]

One can use the extra components of the solution object as well as using zip. For example, say the solution type holds du, the derivative at each timestep. One can comprehend over the values using:

[t+3u-du for (t,u,du) in zip(sol.t,sol.u,sol.du)]

Note that the solution object acts as a vector in time, and so its length is the number of saved timepoints.

## Special Fields

The solution interface also includes some special fields. The problem object prob and the algorithm used to solve the problem alg are included in the solution. Additionally, the field dense is a boolean which states whether the interpolation functionality is available. Further, the field destats contains the internal statistics for the solution process such as the number of linear solves and convergence failures. Lastly, there is a mutable state tslocation which controls the plot recipe behavior. By default, tslocation=0. Its values have different meanings between partial and ordinary differential equations:

• tslocation=0 for non-spatial problems (ODEs) means that the plot recipe will plot the full solution. tslocation=i means that it will only plot the timepoint i.
• tslocation=0 for spatial problems (PDEs) means the plot recipe will plot the final timepoint. tslocation=i means that the plot recipe will plot the ith timepoint.

What this means is that for ODEs, the plots will default to the full plot and PDEs will default to plotting the surface at the final timepoint. The iterator interface simply iterates the value of tslocation, and the animate function iterates the solution calling solve at each step.

## Differential Equation Solver Statistics (destats)

DiffEqBase.DEStatsType
mutable struct DEStats

Statistics from the differential equation solver about the solution process.

Fields

• nf: Number of function evaluations. If the differential equation is a split function, such as a SplitFunction for implicit-explicit (IMEX) integration, then nf is the number of function evaluations for the first function (the implicit function)
• nf2: If the differential equation is a split function, such as a SplitFunction for implicit-explicit (IMEX) integration, then nf2 is the number of function evaluations for the second function, i.e. the function treated explicitly. Otherwise it is zero.
• nw: The number of W=I-gamma*J (or W=I/gamma-J) matrices constructed during the solving process.
• nsolve: The number of linear solves W required for the integration.
• njacs: Number of Jacobians calculated during the integration.
• nnonliniter: Total number of iterations for the nonlinear solvers.
• nnonlinconvfail: Number of nonlinear solver convergence failures.
• ncondition: Number of calls to the condition function for callbacks.
• naccept: Number of accepted steps.
• nreject: Number of rejected steps.
• maxeig: Maximum eigenvalue over the solution. This is only computed if the method is an auto-switching algorithm.

## Return Codes (RetCodes)

The solution types have a retcode field which returns a symbol signifying the error state of the solution. The retcodes are as follows:

• :Default: The solver did not set retcodes.
• :Success: The integration completed without erroring or the steady state solver from SteadyStateDiffEq found the steady state.
• :Terminated: The integration is terminated with terminate!(integrator). Note that this may occur by using TerminateSteadyState from the callback library DiffEqCallbacks.
• :MaxIters: The integration exited early because it reached its maximum number of iterations.
• :DtLessThanMin: The timestep method chose a stepsize which is smaller than the allowed minimum timestep, and exited early.
• :Unstable: The solver detected that the solution was unstable and exited early.
• :InitialFailure: The DAE solver could not find consistent initial conditions.
• :ConvergenceFailure: The internal implicit solvers failed to converge.
• :Failure: General uncategorized failures or errors.

## Problem-Specific Features

Extra fields for solutions of specific problems are specified in the appropriate problem definition page.