Diffusive relaxation systems provide a general framework to approximate nonlinear diffusion problems, also in the degenerate case (Aregba-Driollet et al. in Math. Comput. 73(245):63-94, 2004; Boscarino et al. in Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, 2011; Cavalli et al. in SIAM J. Sci. Comput. 34:A137-A160, 2012; SIAM J. Numer. Anal. 45(5):2098-2119, 2007; Naldi and Pareschi in SIAM J. Numer. Anal. 37:1246-1270, 2000; Naldi et al. in Surveys Math. Indust. 10(4):315-343, 2002). Their discretization is usually obtained by explicit schemes in time coupled with a suitable method in space, which inherits the standard stability parabolic constraint. In this paper we combine the effectiveness of the relaxation systems with the computational efficiency and robustness of the implicit approximations, avoiding the need to resolve nonlinear problems and avoiding stability constraints on time step. In particular we consider an implicit scheme for the whole relaxation system except for the nonlinear source term, which is treated though a suitable linearization technique. We give some theoretical stability results in a particular case of linearization and we provide insight on the general case. Several numerical simulations confirm the theoretical results and give evidence of the stability and convergence also in the case of nonlinear degenerate diffusion.

Cavalli, F., Linearly Implicit Approximations of Diffusive Relaxation Systems, <<ACTA APPLICANDAE MATHEMATICAE>>, 2013; 125 (1): 79-103. [doi:10.1007/s10440-012-9781-4] [http://hdl.handle.net/10807/85481]

Linearly Implicit Approximations of Diffusive Relaxation Systems

Cavalli
Primo
2012

Abstract

Diffusive relaxation systems provide a general framework to approximate nonlinear diffusion problems, also in the degenerate case (Aregba-Driollet et al. in Math. Comput. 73(245):63-94, 2004; Boscarino et al. in Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, 2011; Cavalli et al. in SIAM J. Sci. Comput. 34:A137-A160, 2012; SIAM J. Numer. Anal. 45(5):2098-2119, 2007; Naldi and Pareschi in SIAM J. Numer. Anal. 37:1246-1270, 2000; Naldi et al. in Surveys Math. Indust. 10(4):315-343, 2002). Their discretization is usually obtained by explicit schemes in time coupled with a suitable method in space, which inherits the standard stability parabolic constraint. In this paper we combine the effectiveness of the relaxation systems with the computational efficiency and robustness of the implicit approximations, avoiding the need to resolve nonlinear problems and avoiding stability constraints on time step. In particular we consider an implicit scheme for the whole relaxation system except for the nonlinear source term, which is treated though a suitable linearization technique. We give some theoretical stability results in a particular case of linearization and we provide insight on the general case. Several numerical simulations confirm the theoretical results and give evidence of the stability and convergence also in the case of nonlinear degenerate diffusion.
Inglese
Cavalli, F., Linearly Implicit Approximations of Diffusive Relaxation Systems, <<ACTA APPLICANDAE MATHEMATICAE>>, 2013; 125 (1): 79-103. [doi:10.1007/s10440-012-9781-4] [http://hdl.handle.net/10807/85481]
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/10807/85481
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 1
  • ???jsp.display-item.citation.isi??? ND
social impact