A comparison of adaptive software for 1D parabolic PDEs

In this paper we describe BACOL, a high-order, spatially and temporally adaptive software package for solving systems of one-dimensional parabolic partial differential equations and then compare it with several related software packages. BACOL employs collocation at Gaussian points with a B-spline basis for the spatial discretization. A modification of DASSL is used for the time integration of the resulting differential-algebraic equations. An equidistribution principle is implemented for the spatial mesh adaptation based on a high-quality a posteriori error estimate, and the spatial error tolerance is coupled with the temporal error tolerance to provide a balanced spatial–temporal error control. We compare BACOL with a related software package, EPDCOL, which uses a fixed-spatial-mesh approach, with several other packages which provide spatial and temporal adaptivity, namely D03PPF, TOMS731, MOVCOL, and with one package, HPNEW, which provides spatial and temporal error control. Numerical results demonstrate that BACOL is robust and that it is generally significantly more efficient than existing solvers for problems having solutions with rapid spatial variation.