We present an adapted numerical method for the approximate solution of advection-reaction-diffusion problems on a bidimensional spatial domain and generating periodic wavefronts [3]. In particular, we propose to merge into the numerical scheme the a-priori known information about the qualitative behaviour of the exact solution and the structure of the problem. Traditional finite difference methods could impose a severe reduction of the stepsize in order to accurately follow the oscillations because they are developed in order to achieve exactness (within round-off error) on algebraic polynomials up to a certain degree. Extending the ideas described in [1,2], we develop an adapted method of lines based on trigonometrically fitted finite differences, whose coefficients depend on unknown parameters characterising the exact solution. We deal with the more challenging issue of estimating these parameters by properly manipulating the leading term of the local truncation error a-priori. The vector field of the resulting system of ordinary differential equations is composed by stiff and non-stiff terms, so we suggest to employ an implicit-explicit (IMEX) time method, which implicitly integrates only stiff components and explicitly integrates the others, obtaining advantages in terms of efficiency and stability. The stability and accuracy properties of the overall scheme are rigorously investigated and some numerical tests are presented to show its effectiveness. This is a joint work with Raffaele D’Ambrosio from University of L’Aquila and Beatrice Paternoster from University of Salerno. [1] D’Ambrosio, R., Moccaldi, M., Paternoster, B., Adapted numerical methods for advection-reaction- diffusion problems generating periodic wavefronts, Comput. Math. Appl. 74(5), 1029–1042, 2017. [2] D’Ambrosio, R., Moccaldi, M., Paternoster, B., Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction-diffusion problems, Comput. Phys. Commun. 226, 55–66, 2018. [3] Hundsdorfer, W., Verwer, J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer-Verlag, 2003.

Adapted numerical approximation of advection-reaction-diffusion problems

Moccaldi, Martina;Paternoster, Beatrice
2018

Abstract

We present an adapted numerical method for the approximate solution of advection-reaction-diffusion problems on a bidimensional spatial domain and generating periodic wavefronts [3]. In particular, we propose to merge into the numerical scheme the a-priori known information about the qualitative behaviour of the exact solution and the structure of the problem. Traditional finite difference methods could impose a severe reduction of the stepsize in order to accurately follow the oscillations because they are developed in order to achieve exactness (within round-off error) on algebraic polynomials up to a certain degree. Extending the ideas described in [1,2], we develop an adapted method of lines based on trigonometrically fitted finite differences, whose coefficients depend on unknown parameters characterising the exact solution. We deal with the more challenging issue of estimating these parameters by properly manipulating the leading term of the local truncation error a-priori. The vector field of the resulting system of ordinary differential equations is composed by stiff and non-stiff terms, so we suggest to employ an implicit-explicit (IMEX) time method, which implicitly integrates only stiff components and explicitly integrates the others, obtaining advantages in terms of efficiency and stability. The stability and accuracy properties of the overall scheme are rigorously investigated and some numerical tests are presented to show its effectiveness. This is a joint work with Raffaele D’Ambrosio from University of L’Aquila and Beatrice Paternoster from University of Salerno. [1] D’Ambrosio, R., Moccaldi, M., Paternoster, B., Adapted numerical methods for advection-reaction- diffusion problems generating periodic wavefronts, Comput. Math. Appl. 74(5), 1029–1042, 2017. [2] D’Ambrosio, R., Moccaldi, M., Paternoster, B., Parameter estimation in IMEX-trigonometrically fitted methods for the numerical solution of reaction-diffusion problems, Comput. Phys. Commun. 226, 55–66, 2018. [3] Hundsdorfer, W., Verwer, J., Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer-Verlag, 2003.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/4715346
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