An efficient asymptotic-numerical method to solve nonlinear systems of one-dimensional balance laws
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An efficient asymptotic-numerical method to solve nonlinear systems of one-dimensional balance laws Danilo Costarelli1
· Renato Spigler2
Received: 23 October 2017 / Revised: 29 June 2018 / Accepted: 6 July 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract An asymptotic-numerical method to solve the initial-boundary value problems for systems of balance laws in one space dimension, on the half space is developed. Expansions in powers of t −1/2 are used, in view of the precise asymptotic behavior recently established on theoretical bases. This approach increases considerably the efficiency of a previous one, where just expansions in inverse powers of t were made. Numerical examples and comparisons with the Godunov, the asymptotic high order, and the asymptotic-numerical method earlier developed are presented. Expanding the solution in powers of t −1/2 instead of t −1 , a saving of about one-half of the CPU time can be realized, still achieving the same accuracy. Keywords Systems of balance laws · Dissipative balance laws · Asymptotic-numerical methods Mathematics Subject Classification 41A30 · 65M25 · 35L04 · 35L65
1 Introduction In Costarelli et al. (2013), an asymptotic-numerical method (say ANM-t, for short) was introduced, based on the asymptotic expansion of the solution of Initial-Boundary Value problems for balance laws, in powers of t −1 . The accuracy of such a method increases when the solution is considered for large times. The behavior of the numerical solutions essentially coincides with that obtained by the fourth-order method called “AHO4” (asymptotic high order), due to Briani and Natalini (2006) and Aregba-Driollet et al. (2008). The latter scheme
Communicated by Corina Giurgea.
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Danilo Costarelli [email protected] Renato Spigler [email protected]
1
Department of Mathematics and Computer Sciences, University of Perugia, 1, Via Vanvitelli, 06123 Perugia, Italy
2
Department of Mathematics and Physics, Roma Tre University, 1, Largo S. Leonardo Murialdo, 00146 Rome, Italy
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D. Costarelli, R. Spigler
consists in solving the corresponding stationary equation and then to construct a high-order accurate method with respect to the local truncation error. According to the authors, the AHOp methods (of oder p) can be applied even to dissipative hyperbolic systems arising in a number of applications, but the construction of the corresponding schemes can be difficult. For instance, an application to a quasilinear hyperbolic model of chemotaxis was studied in Natalini and Ribot (1996) and Natalini et al. (2012). The expansion of the solutions in powers of t −1 considered in Costarelli et al. (2013) suggests that the error made when the stationary solution is computed, which might be of order O(t −1 ). On the other hand, Bianchini et al. (2007) proved that, under rather general assumptions, the solution to systems of dissipative balance/conservation laws, in one space dimension, decays when t → +∞, in the L ∞ norm, as t −1/2 ; see also, e.g., C
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