Common Solver Options

Common Solver Options

The DifferentialEquations.jl universe has a large set of common arguments available for the `solve` function. These arguments apply to `solve` on any problem type and are only limited by limitations of the specific implementations.

Many of the defaults depend on the algorithm or the package the algorithm derives from. Not all of the interface is provided by every algorithm. For more detailed information on the defaults and the available options for specific algorithms / packages, see the manual pages for the solvers of specific problems. To see whether a specific package is compaible with the use of a given option, see the Solver Compatibility Chart

Default Algorithm Hinting

To help choose the default algorithm, the keyword argument `alg_hints` is provided to `solve`. `alg_hints` is a `Vector{Symbol}` which describe the problem at a high level to the solver. The options are:

• `:nonstiff` vs `:stiff` - Denotes the equation as nonstiff/stiff.

Currently unused options include:

• `:interpolant` - Denotes that a high-precision interpolation is important.

• `:memorybound` - Denotes that the solver will be memory bound.

This functionality is derived via the benchmarks in DiffEqBenchmarks.jl and is under active development.

SDE Specific

• `:additive` - Denotes that the underlying SDE has additive noise.

• `:stratonovich` - Denotes that the solution should adhere to the Stratonovich interpretation.

Output Control

These arguments control the output behavior of the solvers. It defaults to maximum output to give the best interactive user experience, but can be reduced all the way to only saving the solution at the final timepoint.

The following options are all related to output control. See the "Examples" section at the end of this page for some example usage.

• `dense`: Denotes whether to save the extra pieces required for dense (continuous) output. Default is true for algorithms which have the ability to produce dense output. If dense is false, the solution still acts like a function, and `sol(t)` is a linear interpolation between the saved time points.

• `saveat`: Denotes specific times to save the solution at, during the solving phase. The solver will save at each of the timepoints in this array in the most efficient manner (always including the points of `tspan`). Note that this can be used even if `dense=false`. In fact, if only `saveat` is given, then the arguments `save_everystep` and `dense` are becoming `false` by default and must be explicitly given as `true` if desired. If `saveat` is given a number, then it will automatically expand to `tspan[1]:saveat:tspan[2]`. For methods where interpolation is not possible, `saveat` may be equivalent to `tstops`. Default is `[]`.

• `save_idxs`: Denotes the indices for the components of the equation to save. Defaults to saving all indices. For example, if you are solving a 3-dimensional ODE, and given `save_idxs = [1, 3]`, only the first and third components of the solution will be outputted. Notice that of course in this case the outputed solution will be two-dimensional.

• `tstops`: Denotes extra times that the timestepping algorithm must step to. This should be used to help the solver deal with discontinuities and singularities, since stepping exactly at the time of the discontinuity will improve accuracy. If a method cannot change timesteps (fixed timestep multistep methods), then `tstops` will use an interpolation, matching the behavior of `saveat`. If a method cannot change timesteps and also cannot interpolate, then `tstops` must be a multiple of `dt` or else an error will be thrown. Default is `[]`.

• `d_discontinuities:` Denotes locations of discontinuities in low order derivatives. This will force FSAL algorithms which assume derivative continuity to re-evaluate the derivatives at the point of discontinuity. The default is `[]`.

• `save_everystep`: Saves the result at every timeseries_steps iteration. Default is true if `isempty(saveat)`.

• `timeseries_steps`: Denotes how many steps between saving a value for the timeseries. These "steps" are the steps that the solver stops internally (the ones you get by `save_everystep = true`), not the ones that are instructed by the user (all solvers work in a step-like manner). Defaults to 1.

• `save_start`: Denotes whether the initial condition should be included in the solution type as the first timepoint. Defaults to `true`.

• `save_end`: Denotes whether the final timepoint is forced to be saved, regardless of the other saving settings. Defaults to `true`.

• `initialize_save`: Denotes whether to save after the callback initialization phase (when `u_modified=true`). Defaults to `true`.

Note that `dense` requires `save_everystep=true` and `saveat=false`. If you need additional saving while keeping dense output, see the SavingCallback in the Callback Library.

Stepsize Control

These arguments control the timestepping routines.

Basic Stepsize Control

These are the standard options for controlling stepping behavior. Error estimates do the comparison

\[err_{scaled} = err/(abstol + max(uprev,u)*reltol)\]

The scaled error is guaranteed to be `<1` for a given local error estimate (note: error estimates are local unless the method specifies otherwise). `abstol` controls the non-scaling error and thus can be though of as the error around zero. `reltol` scales with the size of the dependent variables and so one can interpret `reltol=1e-3` as roughly being (locally) correct to 3 digits. Note tolerances can be specified element-wise by passing a vector whose size matches `u0`.

• `adaptive`: Turns on adaptive timestepping for appropriate methods. Default is true.

• `abstol`: Absolute tolerance in adaptive timestepping. This is the tolerance on local error estimatoes, not necessarily the global error (though these quantities are related). Defaults to `1e-6` on deterministic equations (ODEs/DDEs/DAEs) and `1e-2` on stochastic equations (SDEs/RODEs).

• `reltol`: Relative tolerance in adaptive timestepping. This is the tolerance on local error estimatoes, not necessarily the global error (though these quantities are related). Defaults to `1e-3` on deterministic equations (ODEs/DDEs/DAEs) and `1e-2` on stochastic equations (SDEs/RODEs).

• `dt`: Sets the initial stepsize. This is also the stepsize for fixed timestep methods. Defaults to an automatic choice if the method is adaptive.

• `dtmax`: Maximum dt for adaptive timestepping. Defaults are package-dependent.

• `dtmin`: Minimum dt for adaptive timestepping. Defaults are package-dependent.

• `force_dtmin`: Declares whether to continue, forcing the minimum `dt` usage. Default is `false`, which has the solver throw a warning and exit early when encountering the minimum `dt`. Setting this true allows the solver to continue, never letting `dt` go below `dtmin` (and ignoring error tolerances in those cases). Note that `true` is not compatible with most interop packages.

Fixed Stepsize Usage

Note that if a method does not have adaptivity, the following rules apply:

• If `dt` is set, then the algorithm will step with size `dt` each iteration.

• If `tstops` and `dt` are both set, then the algorithm will step with either a size `dt`, or use a smaller step to hit the `tstops` point.

• If `tstops` is set without `dt`, then the algorithm will step directly to each value in `tstops`

• If neither `dt` nor `tstops` are set, the solver will throw an error.

These arguments control more advanced parts of the internals of adaptive timestepping and are mostly used to make it more efficient on specific problems. For detained explanations of the timestepping algorithms, see the timestepping descriptions

• `internalnorm`: The norm function `internalnorm(u)` which error estimates are calculated. Defaults are package-dependent.

• `gamma`: The risk-factor γ in the q equation for adaptive timestepping. Default is algorithm dependent.

• `beta1`: The Lund stabilization α parameter. Defaults are algorithm-dependent.

• `beta2`: The Lund stabilization β parameter. Defaults are algorithm-dependent.

• `qmax`: Defines the maximum value possible for the adaptive q. Defaults are algorithm-dependent.

• `qmin`: Defines the maximum value possible for the adaptive q. Defaults are algorithm-dependent.

• `qsteady_min`: Defines the minimum for the range around 1 where the timestep is held constant. Defaults are algorithm-dependent.

• `qsteady_max`: Defines the maximum for the range around 1 where the timestep is held constant. Defaults are algorithm-dependent.

• `qoldinit`: The initial `qold` in stabilization stepping. Defaults are algorithm-dependent.

• `failfactor`: The amount to decrease the timestep by if the Newton iterations of an implicit method fail. Default is 2.

Miscellaneous

• `maxiters`: Maximum number of iterations before stopping. Defaults to 1e5.

• `callback`: Specifies a callback. Defaults to a callback function which performs the saving routine. For more information, see the Event Handling and Callback Functions manual page.

• `isoutofdomain`: Specifies a function `isoutofdomain(u,p,t)` where, when it returns true, it will reject the timestep. Disabled by default.

• `unstable_check`: Specifies a function `unstable_check(dt,u,p,t)` where, when it returns true, it will cause the solver to exit and throw a warning. Defaults to `any(isnan,u)`, i.e. checking if any value is a NaN.

• `verbose`: Toggles whether warnings are thrown when the solver exits early. Defualts to true.

• `calck`: Turns on and off the internal ability for intermediate interpolations (also known as intermediate density). Not the same as `dense`, which is post-solution interpolation. This defaults to `dense || !isempty(saveat) || "no custom callback is given"`. This can be used to turn off interpolations (to save memory) if one isn't using interpolations when a custom callback is used. Another case where this may be used is to turn on interpolations for usage in the integrator interface even when interpolations are used nowhere else. Note that this is only required if the algorithm doesn't have a free or lazy interpolation (`DP8()`). If `calck = false`, `saveat` cannot be used. The rare keyword `calck` can be useful in event handling.

Progress Monitoring

These arguments control the usage of the progressbar in the Juno IDE.

• `progress`: Turns on/off the Juno progressbar. Default is false.

• `progress_steps`: Numbers of steps between updates of the progress bar. Default is 1000.

• `progress_name`: Controls the name of the progressbar. Default is the name of the problem type.

• `progress_message`: Controls the message with the progressbar. Defaults to showing `dt`, `t`, the maximum of `u`.

User Data

• `userdata`: This is a user-chosen type which will show up in the `integrator` type, allowing the user to have a cache for callbacks, event handling, and other various activities.

Error Calculations

If you are using the test problems (ex: `ODETestProblem`), then the following options control the errors which are calculated:

• `timeseries_errors`: Turns on and off the calculation of errors at the steps which were taken, such as the `l2` error. Default is true.

• `dense_errors`: Turns on and off the calculation of errors at the steps which require dense output and calculate the error at 100 evenly-spaced points throughout `tspan`. An example is the `L2` error. Default is false.

Examples

The following lines are examples of how one could use the configuration of `solve()`. For these examples a 3-dimensional ODE problem is assumed, however the extention to other types is straightforward.

1. `solve(prob, AlgorithmName())` : The "default" setting, with a user-specified

algorithm (given by `AlgorithmName()`).All parameters get their default values. This means that the solution is saved at the steps the Algorithm stops internally and dense output is enabled if the chosen algorithm allows for it. All other integration parameters (e.g. stepsize) are chosen automatically.

1. `solve(prob, saveat = 0.01, abstol = 1e-9, reltol = 1e-9)` : Standard setting

for accurate output at specified (and equidistant) time intervals, used for e.g. Fourier Transform. The solution is given every 0.01 time units, starting from `tspan[1]`. The solver used is Tsit5() since no keyword `alg_hits` is given.

1. `solve(prob, maxiters = 1e7, progress = true, save_idxs = [1])` : Using longer

maximum number of solver iterations can be useful when a given `tspan` is very long. This example only saves the first of the variables of the system, either to save size or because the user does not care about the others. Finally, with `progress = true` you are enabling the progress bar, provided you are using the Atom+Juno IDE set-up for your Julia.