# Convergence Simulations

The convergence simulation type is useful for deriving order of convergence estimates from a group of simulations. This object will automatically assemble error vectors into a more useful manner and provide plotting functionality. Convergence estimates are also given by pair-wise estimates.

One can automatically have DifferentialEquations.jl perform the error analysis by passing a ConvergenceSimulation a vector of solutions, or using one of the provided test_convergence functions. These will give order of convergence estimates and provide plotting functionality. This requires that the true solution was provided in the problem definition.

ConvergenceSimulations can either be created by passing the constructor the appropriate solution array or by using one of the provided test_convergence functions.

## The ConvergenceSimulation Type

A type which holds the data from a convergence simulation.

### Fields

• solutions::Array{<:DESolution}: Holds all the PdeSolutions.

• errors: Dictionary of the error calculations. Can contain:

• h1Errors: Vector of the H1 errors.
• l2Errors: Vector of the L2 errors.
• maxErrors: Vector of the nodal maximum errors.
• node2Errors: Vector of the nodal l2 errors.
• N: The number of simulations.

• auxdata: Auxillary data of the convergence simluation. Entries can include:

• dts: The dt's in the simulations.
• dxs: The dx's in the simulations.
• μs: The CFL μ's in the simulations.
• νs: The CFL ν's in the simulations.
• 𝒪est: Dictionary of order estimates. Can contain:

• ConvEst_h1: The H1 error order of convergence estimate for the convergence simulation. Generated via log2(error[i+1]/error[i]). Thus only valid if generated by halving/doubling the dt/dx. If alternate scaling, modify by dividing of log(base,ConvEst_h1)
• ConvEst_l2: The L2 error order of convergence estimate for the convergence simulation. Generated via log2(error[i+1]/error[i]). Thus only valid if generated by halving/doubling the dt/dx. If alternate scaling, modify by dividing of log(base,ConvEst_l2)
• ConvEst_max: The nodal maximum error order of convergence estimate for the convergence simulation. Generated via log2(error[i+1]/error[i]). Thus only valid if generated by halving/doubling the dt/dx. If alternate scaling, modify by dividing of log(base,ConvEst_max)
• ConvEst_node2: The nodal l2 error order of convergence estimate for the convergence simulation. Generated via log2(error[i+1]/error[i]). Thus only valid if generated by halving/doubling the dt/dx. If alternate scaling, modify by dividing of log(base,ConvEst_node2)
• convergence_axis: The axis along which convergence is calculated. For example, if we calculate the dt convergence, convergence_axis is the dts used in the calculation.

## Plot Functions

The plot functionality is provided by a Plots.jl recipe. What is plotted is a line series for each calculated error along the convergence axis. To plot a convergence simulation, simply use:

plot(sim::ConvergenceSimulation)

All of the functionality (keyword arguments) provided by Plots.jl are able to be used in this command. Please see the Plots.jl documentation for more information.

## ODE

test_convergence(dts::AbstractArray,prob::AbstractODEProblem)

Tests the order of the time convergence of the given algorithm on the given problem solved over the given dts. Keyword arguments are passed to the ODE solver.

## SDE

test_convergence(dts::AbstractArray,prob::AbstractSDEProblem)

Tests the strong order time convergence of the given algorithm on the given problem solved over the given dts. Keyword arguments are passed to the ODE solver. Except:

• numMonte: The number of simulations for each dt. Default is 10000.

### Order Estimation

calc𝒪estimates(error::Vector{Number})`

Computes the pairwise convergence estimate for a convergence test done by halving/doubling stepsizes via

log2(error[i+1]/error[i])

Returns the mean of the convergence estimates.