Our aim is to solve systems of ordinary differential equations potentially candidate to be stiff, so developed methods must have high stability properties and uniform order of convergence. We focus our attention on multivalue methods [1], which are a generalization of classical methods, such as multistep and Runge-Kutta methods, and we extend the solution smoothly by approximating them though a collocation polynomial. These methods require at each time-step the solution of a non linear system of internal stages, so the computational effort is strictly connected to the nature of this system. We are interested in the construction of methods that allow a reduction of this computational cost, so we propose methods with full matrix [6] and structured matrix (triangular [2], singly triangular [3, 4], diagonal [5]). In the case of structured matrix, we perform almost collocation since it is not possible to impose all the collocation conditions. We compare these methods calculating the error in the final step point and the experimental order of convergence. References [1] J.C. Butcher, General linear methods. Computers & Mathematics with Applications. 31 (4-5): 105-112. doi:10.1016/0898-1221(95)00222-7 (1996). [2] D. Conte, R. D'Amborsio, M.P. D'Arienzo, B. Paternoster, Highly stable multivalue collocation methods, J. Phys.: Conf. Ser. 1564, 012012 (2020). [3] D. Conte, R. D'Ambrosio, M.P. D'Arienzo, B. Paternoster, Singly diagonally implicit multivalue collocation methods, in 2020 International Conference on Mathematics and Computers in Science and Engineering (MACISE), Madrid, Spain, DOI:10.1109/MACISE49704.2020.00018, pp. 65-68 (2020). [4] D. Conte, R. D'Amborsio, M.P. D'Arienzo, B. Paternoster, One-point spectrum Nordsieck almost collocation methods, International Journal of Circuits, Systems and Signal Processing, vol. 14, pag. 266-275, DOI: 10.46300/9106.2020.14.38 (2020). [5] D. Conte, R. D'Amborsio, M.P. D'Arienzo, B. Paternoster, Multivalue almost collocation methods with diagonal coecient matrix, Lecture Notes in Comput. Sci., in press. [6] R. D'Ambrosio, B. Paternoster, Multivalue collocation methods free from order reduction,J. Comput. Appl. Math. DOI: 10.1016/j.cam.2019.112515 (2019).

Highly Stable Multivalue Almost Collocation Methods with Structured Coefficient Matrix

Maria Pia D'Arienzo
;
Dajana Conte;Beatrice Paternoster
2020-01-01

Abstract

Our aim is to solve systems of ordinary differential equations potentially candidate to be stiff, so developed methods must have high stability properties and uniform order of convergence. We focus our attention on multivalue methods [1], which are a generalization of classical methods, such as multistep and Runge-Kutta methods, and we extend the solution smoothly by approximating them though a collocation polynomial. These methods require at each time-step the solution of a non linear system of internal stages, so the computational effort is strictly connected to the nature of this system. We are interested in the construction of methods that allow a reduction of this computational cost, so we propose methods with full matrix [6] and structured matrix (triangular [2], singly triangular [3, 4], diagonal [5]). In the case of structured matrix, we perform almost collocation since it is not possible to impose all the collocation conditions. We compare these methods calculating the error in the final step point and the experimental order of convergence. References [1] J.C. Butcher, General linear methods. Computers & Mathematics with Applications. 31 (4-5): 105-112. doi:10.1016/0898-1221(95)00222-7 (1996). [2] D. Conte, R. D'Amborsio, M.P. D'Arienzo, B. Paternoster, Highly stable multivalue collocation methods, J. Phys.: Conf. Ser. 1564, 012012 (2020). [3] D. Conte, R. D'Ambrosio, M.P. D'Arienzo, B. Paternoster, Singly diagonally implicit multivalue collocation methods, in 2020 International Conference on Mathematics and Computers in Science and Engineering (MACISE), Madrid, Spain, DOI:10.1109/MACISE49704.2020.00018, pp. 65-68 (2020). [4] D. Conte, R. D'Amborsio, M.P. D'Arienzo, B. Paternoster, One-point spectrum Nordsieck almost collocation methods, International Journal of Circuits, Systems and Signal Processing, vol. 14, pag. 266-275, DOI: 10.46300/9106.2020.14.38 (2020). [5] D. Conte, R. D'Amborsio, M.P. D'Arienzo, B. Paternoster, Multivalue almost collocation methods with diagonal coecient matrix, Lecture Notes in Comput. Sci., in press. [6] R. D'Ambrosio, B. Paternoster, Multivalue collocation methods free from order reduction,J. Comput. Appl. Math. DOI: 10.1016/j.cam.2019.112515 (2019).
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11386/4755340
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact