Error-control B-spline Gaussian collocation PDE software with event detection

Jack Pew; Connor Tannahill; Paul Muir; Jack Pew; Connor Tannahill; Paul Muir

doi:10.3934/math.2026131

AIMS Mathematics

2026, Volume 11, Issue 2: 3243-3268. doi: 10.3934/math.2026131

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Error-control B-spline Gaussian collocation PDE software with event detection

Saint Mary's University, Halifax, NS, Canada, B3H 3C3

Received: 17 April 2024 Revised: 14 May 2025 Accepted: 23 January 2026 Published: 03 February 2026
MSC : 65L06, 65L10, 65L80, 65M20, 65M70

This paper introduces BACOLIKR, a new software package for the error-controlled numerical solution of systems of one-dimensional time-dependent partial differential equations (PDEs). A novel feature of this package is that it allows the user to specify a solution dependent condition, called an event, and then the software will determine the point in time at which the specified event occurs. This event detection capability can be used to provide an efficient and accurate means for dealing with time-dependent discontinuities in the PDEs or the boundary conditions.
BACOLIKR employs adaptive B-spline Gaussian collocation for the spatial discretization of the PDE system within a spatial error control algorithm. The event detection capability in BACOLIKR is based on its use of a modified version of the time integrator, DASKR, which implements event detection for time-dependent differential-algebraic equations. BACOLIKR was developed through modifications of an earlier error control PDE solver, BACOLI.
In this paper, we provide an overview of the BACOLI and DASKR packages and then describe the modifications that were made in order to develop BACOLIKR. We then show how BACOLIKR can be used for the effective solution of a number of PDE-based event detection problems including solution layer-boundary intersection detection and solution layer merge detection in a fluid mechanics model, critical tumor mass detection in a brain tumor model, steady state detection in the Gierer-Meinhardt model, and boundary condition event detection in a discontinuous heat flow model.
- B-splines,
- collocation,
- interpolation,
- error estimation,
- error control,
- event detection,
- partial differential equations (PDEs)
Citation: Jack Pew, Connor Tannahill, Paul Muir. Error-control B-spline Gaussian collocation PDE software with event detection[J]. AIMS Mathematics, 2026, 11(2): 3243-3268. doi: 10.3934/math.2026131

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Abstract

This paper introduces BACOLIKR, a new software package for the error-controlled numerical solution of systems of one-dimensional time-dependent partial differential equations (PDEs). A novel feature of this package is that it allows the user to specify a solution dependent condition, called an event, and then the software will determine the point in time at which the specified event occurs. This event detection capability can be used to provide an efficient and accurate means for dealing with time-dependent discontinuities in the PDEs or the boundary conditions.

BACOLIKR employs adaptive B-spline Gaussian collocation for the spatial discretization of the PDE system within a spatial error control algorithm. The event detection capability in BACOLIKR is based on its use of a modified version of the time integrator, DASKR, which implements event detection for time-dependent differential-algebraic equations. BACOLIKR was developed through modifications of an earlier error control PDE solver, BACOLI.

In this paper, we provide an overview of the BACOLI and DASKR packages and then describe the modifications that were made in order to develop BACOLIKR. We then show how BACOLIKR can be used for the effective solution of a number of PDE-based event detection problems including solution layer-boundary intersection detection and solution layer merge detection in a fluid mechanics model, critical tumor mass detection in a brain tumor model, steady state detection in the Gierer-Meinhardt model, and boundary condition event detection in a discontinuous heat flow model.

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