Speaker
Description
Parabolic PDEs are widely used to model phenomena evolving in time and space, but their numerical solution requires efficient time integration combined with suitable spatial discretizations.
Thus, we investigate approaches to improve the efficiency of pre-existing methods used as time integrators for the ODE system arising from the spatial semi-discretization of the given PDE. Considering W methods of order up to four [1,4,5] as underlying schemes, we compare the use of AMF (Approximate Matrix Factorization) and matrix-oriented techniques. We show that, by exploiting the structure of the matrices arising from the spatial discretization of the diffusion operator, these approaches significantly reduce computational cost, while preserving the desired stability and accuracy properties [2].
Second, we investigate the improvement of the accuracy of the AMF-W methods when PDEs with time-dependent boundary conditions are considered. In such cases, AMF schemes may suffer from order reduction. Thus, we will exploit the application of boundary correction techniques that aim to recover the expected order of accuracy, exploiting problems with several typologies of boundary conditions in multiple space dimensions [3,4].
References
- D. Conte, S. González-Pinto, D. Hernández-Abreu, and G. Pagano. On approximate matrix factorization and TASE W-methods for the time integration of parabolic partial differential equations. J. Sci. Comput. 100(2), 34. 2024.
- D. Conte, S. Iscaro, G. Pagano. On Matrix-Oriented and AMF-W Methods for advection-reaction-diffusion Partial Differential Equations. In preparation.
- S. González-Pinto, D. Hernández-Abreu, S. Iscaro. On the treatment of boundary conditions for AMF-W methods in diffusion-reaction PDEs. In preparation.
- González-Pinto, S., Hernández-Abreu, D. Boundary corrections for splitting methods in the time integration of multidimensional parabolic problems. Appl. Numer. Math., 210: 95-112. 2025.
- W. Hundsdorfer, J. G. Verwer. Numerical solution of time-dependent advection diffusion-reaction equations. Vol. 33. Springer Science and Business Media., 2003.