The talk focuses on the numerical solution of advection-reaction-diffusion problems by adapted finite difference schemes. In other terms, the numerical scheme is developed in order to exploit the a-priori knowledge of the qualitative behaviour of the solution, gaining advantages in terms of efficiency and accuracy with respect to classical schemes already known in literature, which mostly rely on algebraic polynomials. The adaptation is carried out by the so-called trigonometrical fitting technique for the space-discretization, giving rise to a system of ODEs whose vector field contains both stiff and non-stiff terms. Due to this mixed nature of the vector field, an Implicit-Explicit (IMEX) method is employed for the time-integration. The coefficients of the introduced numerical scheme depend on unknown parameters which have to be properly estimated: such an estimate is performed by an efficient offline minimization of the leading term of the local truncation error. The effectiveness of this problem-oriented approach is shown through a rigorous theoretical analysis and selected numerical experiments. References [1] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. 2017 Adapted numerical methods for advection-reaction-diffusion problems generating periodic wavefronts. Comput. Math. Appl. 74(5), 1029–1042. [2] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. 2018 Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction-diffusion problems., Comput. Phys. Commun. 226, 55–66. [3] Perumpanani, A.J., Sherratt, J.A., Maini, P.K. 1995 Phase differences in reaction–diffusion–advection systems and applications to morphogenesis, J. Appl. Math. 55, 19–33.
Adapted time-integration of partial differential equations generating periodic wavefronts
Moccaldi, Martina;Paternoster, Beatrice
2018-01-01
Abstract
The talk focuses on the numerical solution of advection-reaction-diffusion problems by adapted finite difference schemes. In other terms, the numerical scheme is developed in order to exploit the a-priori knowledge of the qualitative behaviour of the solution, gaining advantages in terms of efficiency and accuracy with respect to classical schemes already known in literature, which mostly rely on algebraic polynomials. The adaptation is carried out by the so-called trigonometrical fitting technique for the space-discretization, giving rise to a system of ODEs whose vector field contains both stiff and non-stiff terms. Due to this mixed nature of the vector field, an Implicit-Explicit (IMEX) method is employed for the time-integration. The coefficients of the introduced numerical scheme depend on unknown parameters which have to be properly estimated: such an estimate is performed by an efficient offline minimization of the leading term of the local truncation error. The effectiveness of this problem-oriented approach is shown through a rigorous theoretical analysis and selected numerical experiments. References [1] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. 2017 Adapted numerical methods for advection-reaction-diffusion problems generating periodic wavefronts. Comput. Math. Appl. 74(5), 1029–1042. [2] D’Ambrosio, R., Moccaldi, M. and Paternoster, B. 2018 Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction-diffusion problems., Comput. Phys. Commun. 226, 55–66. [3] Perumpanani, A.J., Sherratt, J.A., Maini, P.K. 1995 Phase differences in reaction–diffusion–advection systems and applications to morphogenesis, J. Appl. Math. 55, 19–33.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
