This paper presents the culmination of software development for one-dimensional parabolic partial differential equations (PDEs), which many people have contributed to over the past three decades. The described program is based on the use of high quality software, in a combination of different areas that are involved in developing a unified package for solving these one-dimensional problems. These areas include spline representation, ordinary differential equation (ODE) software integration, differential/algebraic methods, and mesh adaptation methods. This combined software package addresses the need for a package that simultaneously deals with spatial and temporal error control and the numerical solution of such problems.

The paper, however, is about more than the software integration (which is well described in this and the companion online appendix). It provides a well-balanced history of development for the nonspecialist, with a derivation of error analysis for temporal and spatial approximation. In order to address issues of performance and efficiency, the authors have modified the differential algebraic system solver (DASSL) software, and exploit the linear algebraic structure of the almost block diagonal matrices involved. In addition to providing numerical examples, the paper has a clear explanation of the software structure and user-supplied routines. The paper concludes with a range of projects for future work.