Cone-grid scheme for solving hyperbolic systems of conservation laws and one application
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Cone-grid scheme for solving hyperbolic systems of conservation laws and one application Mahmoud A. E. Abdelrahman1
Received: 20 May 2017 / Revised: 28 September 2017 / Accepted: 16 October 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract In this work, we introduce a new numerical method, the so-called cone-grid scheme for solving hyperbolic systems of conservation laws. The formulation and the theoretical analysis for this scheme are slightly simpler than other algorithms. Simultaneously, our technique leads to a more efficacious numerical scheme. The Riemann solution is the basic ingredient for this scheme. We also give the main application of this technique, namely the global weak solutions for the one-dimensional model prescribes heat conduction in solids at low temperature, which called phonon–Bose model. This system consists of a conservation equation for the energy density and the heat flux. The scheme also satisfies positivity. The cone-grid scheme was compared with exact solution by three numerical examples, where explicit solutions are known. These numerical results verify the accuracy of the proposed scheme qualitatively and quantitatively. Keywords Hyperbolic conservation laws · Cone-grid scheme · Heat conduction model · Rarefaction waves (fans) · Shock waves · Initial value problems · L 1 -stability Mathematics Subject Classification 35L45 · 35L60 · 35L65 · 35L67 · 65M99
1 Introduction We are concerned with the construction of a new global weak solution to the strictly hyperbolic systems of conservation laws: Wt + F(W )x = 0.
(1.1)
Communicated by Pierangelo Marcati.
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Mahmoud A. E. Abdelrahman [email protected] Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
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M. A. E. Abdelrahman
In Abdelrahman and Kunik (2014), we presented an interesting scheme for the first time, the so-called cone-grid. The very characteristic feature of this scheme is that it is surprisingly implicit, but often algebraically solvable and unconditionally stable, i.e., no Courant–Friedrichs–Lewy (CFL) condition is needed. This condition appears automatically owing to the natural construction of light cones. The unconditionally stable is satisfied only for special systems of conservation laws with a finite speed of propagation, where a natural CFL-condition is used for the proposed numerical scheme. Indeed one of the most important feature, that this is scheme is so easy to implement. In fact all these features are not satisfied for many other schemes, like finite volume (FV), (W)ENO high-order FV, discontinuous Galerkin. The cone scheme is robustly based on the integral conservation laws. Actually this scheme can be used as a box solver for other conservation laws, for example see Abdelrahman and Kunik (2014) and Abdelrahman (2016). Nevertheless, this scheme can be generalized to a general system (1.1). In the first part of this paper we will revisit one-dimensional cone-grid scheme construction, which will be used in the
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