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.

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

sol[i]

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

sol.t[i]

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

sol[i,j]

will be the jth component at timestep i. If the independent variables had shape (for example, was a matrix), then j is the linear index. We can also access solutions with shape:

sol[i,j,k]

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

sol[:,j]

gives the timeseries for the jth component.

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 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 (t,u) 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. 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.

Problem-Specific Features

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