We propose an adapted numerical approximation of advection-reaction-diffusion problems on a bi-dimensional spatial domain and generating periodic wavefronts [3]. The adaptation is carried out by merging into the numerical scheme information known in advance concerning the qualitative behaviour of the exact solution and the structure of the problem. Classic numerical methods based on finite difference formulae could strongly reduce the stepsize in order to accurately reproduce the prescribed oscillations of the exact solution because they are constructed to be exact (within round-off error) on algebraic polynomials up to a certain degree. Broadening the investigation presented in [1, 2], we devise an adapted method of lines combined with trigonometrically-fitted finite differences. However, the coefficients of these formulae rely on unknown parameters, related to the exact solution, which we propose to estimate through an appropriate treatment of the local truncation error. The resulting system of ordinary differential equations exhibits a vector field consisting of stiff and nonstiff terms, so we adopt an implicit-explicit (IMEX) time solver, which implicitly integrates only stiff constituents and explicitly integrates the others, gaining benefits in terms of efficiency and accuracy. A rigorous analysis on the stability and accuracy properties of the overall method is presented, joint with some numerical tests, in order to highlight its effectiveness. 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] Hundsdorfer, W. and Verwer, J. 2003 Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer-Verlag.
Trigonometrically fitted-IMEX discretization of advection-reaction-diffusion problems
Moccaldi, Martina;Paternoster, Beatrice
2018-01-01
Abstract
We propose an adapted numerical approximation of advection-reaction-diffusion problems on a bi-dimensional spatial domain and generating periodic wavefronts [3]. The adaptation is carried out by merging into the numerical scheme information known in advance concerning the qualitative behaviour of the exact solution and the structure of the problem. Classic numerical methods based on finite difference formulae could strongly reduce the stepsize in order to accurately reproduce the prescribed oscillations of the exact solution because they are constructed to be exact (within round-off error) on algebraic polynomials up to a certain degree. Broadening the investigation presented in [1, 2], we devise an adapted method of lines combined with trigonometrically-fitted finite differences. However, the coefficients of these formulae rely on unknown parameters, related to the exact solution, which we propose to estimate through an appropriate treatment of the local truncation error. The resulting system of ordinary differential equations exhibits a vector field consisting of stiff and nonstiff terms, so we adopt an implicit-explicit (IMEX) time solver, which implicitly integrates only stiff constituents and explicitly integrates the others, gaining benefits in terms of efficiency and accuracy. A rigorous analysis on the stability and accuracy properties of the overall method is presented, joint with some numerical tests, in order to highlight its effectiveness. 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] Hundsdorfer, W. and Verwer, J. 2003 Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer-Verlag.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
