A new basis implementation for a mixed order boundary value ODE solver
SIAM Journal on Scientific and Statistical Computing
Error equidistribution and mesh adaptation
SIAM Journal on Scientific Computing
Moving mesh partial differential equations (MMPDES) based on the equidistribution principle
SIAM Journal on Numerical Analysis
Collocation Software for Boundary-Value ODEs
ACM Transactions on Mathematical Software (TOMS)
On the numerical solution of one-dimensional PDEs using adaptive methods based on equidistribution
Journal of Computational Physics
A BVP solver based on residual control and the Maltab PSE
ACM Transactions on Mathematical Software (TOMS)
Variational mesh adaptation: isotropy and equidistribution
Journal of Computational Physics
A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws
SIAM Journal on Scientific Computing
Digital filters in adaptive time-stepping
ACM Transactions on Mathematical Software (TOMS)
Adaptive Mesh Methods for One- and Two-Dimensional Hyperbolic Conservation Laws
SIAM Journal on Numerical Analysis
Time-step selection algorithms: adaptivity, control, and signal processing
Applied Numerical Mathematics - The third international conference on the numerical solutions of volterra and delay equations, May 2004, Tempe, AZ
Efficient Numerical Solution of the Density Profile Equation in Hydrodynamics
Journal of Scientific Computing
Adaptivity and computational complexity in the numerical solution of ODEs
Journal of Complexity
From nonlinear PDEs to singular ODEs
Applied Numerical Mathematics
Efficient mesh selection for collocation methods applied to singular BVPs
Journal of Computational and Applied Mathematics
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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 驴(x). The local mesh width Δx j驴+驴1/2驴=驴x j驴+驴1驴驴驴x j with 0驴=驴x 0驴x 1驴x N 驴=驴1 is computed as Δx j驴+驴1/2驴=驴驴 N / 驴 j驴+驴1/2, where $\{\varphi_{j+1/2}\}_0^{N-1}$ is a discrete approximation to the continuous density function 驴(x), representing mesh width variation. The parameter 驴 N 驴=驴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 驴(x) is determined, another control law determines N based on the prescribed tolerance ${\textsc {tol}}$ . 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.