Automatic grid control in adaptive BVP solvers

• Published in: Numerical algorithms Vol. 56; no. 1; pp. 61 - 92
• Main Authors: Pulverer, Gernot, Söderlind, Gustaf, Weinmüller, Ewa
• Format: Journal Article
• Language: English
• Published: Boston Springer US 2011; Springer Nature B.V
• Subjects: Adaptive control; Adaptivity; Algebra; Algorithms; Automatic control; Boundary value problems; Computer Science; Continuity (mathematics); Control systems; Control theory; Density; Digital filters; Error equidistribution; Errors; Estimates; Feedback control; Grid generation; Grid refinement; Matematik; Mathematical analysis; Mathematical models; Mathematics; Modular systems; Natural Sciences; Naturvetenskap; Numeric Computing; Numerical Analysis; Ordinary differential equations; Original Paper; Singular problems; Singularly perturbed problems; Solvers; Step size control; Theory of Computation; Tolerances; Adaptivity; Singularly perturbed problems; Boundary value problems; Step size control; Grid refinement; Ordinary differential equations; Error equidistribution; Singular problems; Grid generation
• ISSN: 1017-1398; 1572-9265; 1572-9265
• DOI: 10.1007/s11075-010-9374-0

Grid adaptation in two-point boundary value problems is usually based on mapping a uniform auxiliary grid to the desired nonuniform grid. Here we combine this approach with a new control system for constructing a grid density function
ϕ
(
x
). The local mesh width Δ
x
j
 + 1/2
 = 
x
j
 + 1
 − 
x
j
with 0 = 
x
0
 < 
x
1
 < ... < 
x
N
 = 1 is computed as Δ
x
j
 + 1/2
 = 
ε
N
/
φ
j
 + 1/2
, where
is a discrete approximation to the continuous density function
ϕ
(
x
), representing mesh width variation. The parameter
ε
N
 = 1/
N
controls accuracy via the choice of
N
. For any given grid, a solver provides an error estimate. Taking this as its input, the feedback control law then adjusts the grid, and the interaction continues until the error has been equidistributed. Digital filters may be employed to process the error estimate as well as the density to ensure the regularity of the grid. Once
ϕ
(
x
) is determined, another control law determines
N
based on the prescribed tolerance
. The paper focuses on the interaction between control system and solver, and the controller’s ability to produce a near-optimal grid in a stable manner as well as correctly predict how many grid points are needed. Numerical tests demonstrate the advantages of the new control system within the
bvpsuite
solver,
ceteris paribus
, for a selection of problems and over a wide range of tolerances. The control system is modular and can be adapted to other solvers and error criteria.