Error Control Adaptivity

Adaptivity helps ensure the quality of the our numerical solution, and when our solution exhibits significant estimating errors, adaptivity automatically refine the mesh based on the error distribution, and providing a final satisfying solution.

When comes to solving ill-conditioned BVP, for example the singular pertubation problem where the small parameters become extremally small leading to the layers phonemona, the error control adaptivity becomes even more critical, because the minor pertubations can lead to large deviation in the solution. In such cases, adaptivity autimatically figure out where to use refined mesh and where to use coarse mesh to achieve the balance of computational efficiency and accuracy.

BoundaryValuDiffEq.jl support error control adaptivity for collocation methods, and the adaptivity is default as defect control adaptivity when using adaptive collocation solvers:

sol = solve(prob, MIRK4(), dt = 0.01, adaptive = true)

Actually, BoundaryValueDiffEq.jl supports both defect and global error control adaptivity(while the defect control is the default controller) Boisvert et al. [4], to specify different error control metods, we simply need to specify the controller keyword in solve:

sol = solve(prob, MIRK4(), dt = 0.01, controller = GlobalErrorControl()) # Use global error control
sol = solve(prob, MIRK4(), dt = 0.01, controller = SequentialErrorControl()) # Use Sequential error control
sol = solve(prob, MIRK4(), dt = 0.01, controller = HybridErrorControl()) # Use Hybrid error control

Error control methods

BoundaryValueDiffEqCore.DefectControl — Type

DefectControl(; defect_threshold = 0.1)

Defect estimation method with defect defined as

\[defect = \max\frac{S'(x) - f(x,S(x))}{1 + |f(x,S(x))|}\]

Defect controller, with the maximum defect_threshold as 0.1, when the estimating defect is greater than the defect_threshold, the mesh will be refined.

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BoundaryValueDiffEqCore.GlobalErrorControl — Type

GlobalErrorControl(; method = HOErorControl())

Global error controller, use high order global error estimation method HOErrorControl as default.

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BoundaryValueDiffEqCore.SequentialErrorControl — Type

SequentialErrorControl(; defect = DefectControl(), global_error = GlobalErrorControl())

First use the defect controller, if the defect is satisfying, then use global error controller.

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BoundaryValueDiffEqCore.HybridErrorControl — Type

HybridErrorControl(; DE = 1.0, GE = 1.0, defect = DefectControl(), global_error = GlobalErrorControl())

Control both of the defect and global error, where the error norm is the linear combination of the defect and global error.

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BoundaryValueDiffEqCore.NoErrorControl — Type

NoErrorControl()

No error control method.

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While we can achieve global error control in different ways, we can use different methods to estimate the global error:

BoundaryValueDiffEqCore.HOErrorControl — Type

HOErrorControl()

Higher order global error estimation method

Uses a solution from order+2 method on the original mesh and calculate the error with

\[error = \max\frac{u_p - u_{p+2}}{1 + |u_p|}\]

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BoundaryValueDiffEqCore.REErrorControl — Type

REErrorControl()

Richardson extrapolation global error estimation method

Use Richardson extrapolation to calculate the error on the doubled mesh with

\[error = \frac{2^p}{2^p-1} * \max\frac{u_h - u_{h/2}}{1 + |u_h|}\]

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