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Abstract: Focusing on the local geometric properties of the shockpeakon for the Degasperis–Procesi equation, a multi-symplectic method for the quasi-Degasperis–Procesi equation is proposed to reveal the jump discontinuity of the shockpeakon for the Degasperis–Procesi equation numerically in this paper. The main contribution of this paper lies in the following: (1) the uniform multi-symplectic structure of the b-family equation is constructed; (2) the stable jump discontinuity of the shockpeakon for the Degasperis–Procesi equation is reproduced by simulating the peakon–antipeakon collision process of the quasi-Degasperis–Procesi equation. First, the multi-symplectic structure and several local conservation laws are presented for the b-family equation with two exceptions ( and ). And then, the Preissman Box multi-symplectic scheme for the multi-symplectic structure is constructed and the mathematical proofs for the discrete local conservation laws of the multi-symplectic structure are given. Finally, the numerical experiments on the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation are reported to investigate the jump discontinuity of shockpeakon of the Degasperis–Procesi equation. From the numerical results, it can be concluded that the peakon–antipeakon collision of the quasi-Degasperis–Procesi equation can be simulated well by the multi-symplectic method and the simulation results can reveal the jump discontinuity of shockpeakon of the Degasperis–Procesi equation approximately. [Copyright &y& Elsevier]

We consider the relativistic Vlasov–Maxwell (RVM) equations in the limit when the light velocity c goes to infinity. In this regime, the RVM system converges towards the Vlasov–Poisson system and the aim of this paper is to construct asymptotic preserving numerical schemes that are robust with respect to this limit. Our approach relies on a time splitting approach for the RVM system employing an implicit time integrator for Maxwell’s equations in order to damp the higher and higher frequencies present in the numerical solution. A number of numerical simulations are conducted in order to investigate the performances of our numerical scheme both in the relativistic as well as in the classical limit regime. In addition, we derive the dispersion relation of the Weibel instability for the continuous and the discretized problem. [ABSTRACT FROM AUTHOR]

Numerical kinetic models of plasma turbulence require careful treatment of conserved quantities. In the collisionless limit, numerical dissipation can impact entropy in a non-controlled manner. In this paper, the impact of the error in entropy conservation is investigated. In a simulation of ion-acoustic turbulence, a large error ( 15 % ) in entropy conservation is found. Surprisingly, this error is independent of the numerical method, scheme, or number of grid points. Adding a collision operator resolves this issue, but only if the collision frequency is large enough that it modifies the qualitative time evolution of observables, such as electric field amplitude, anomalous resistivity, or phase-space structures. [ABSTRACT FROM AUTHOR]

OpenSMOKE++ is a general framework for numerical simulations of reacting systems with detailed kinetic mechanisms, including thousands of chemical species and reactions. The framework is entirely written in object-oriented C++ and can be easily extended and customized by the user for specific systems, without having to modify the core functionality of the program. The OpenSMOKE++ framework can handle simulations of ideal chemical reactors (plug-flow, batch, and jet stirred reactors), shock-tubes, rapid compression machines, and can be easily incorporated into multi-dimensional CFD codes for the modeling of reacting flows. OpenSMOKE++ provides useful numerical tools such as the sensitivity and rate of production analyses, needed to recognize the main chemical paths and to interpret the numerical results from a kinetic point of view. Since simulations involving large kinetic mechanisms are very time consuming, OpenSMOKE++ adopts advanced numerical techniques able to reduce the computational cost, without sacrificing the accuracy and the robustness of the calculations. In the present paper we give a detailed description of the framework features, the numerical models available, and the implementation of the code. The possibility of coupling the OpenSMOKE++ functionality with existing numerical codes is discussed. The computational performances of the framework are presented, and the capabilities of OpenSMOKE++ in terms of integration of stiff ODE systems are discussed and analyzed with special emphasis. Some examples demonstrating the ability of the OpenSMOKE++ framework to successfully manage large kinetic mechanisms are eventually presented. Program summary Program title: OpenSMOKE++ Catalogue identifier: AEVY_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEVY_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: GNU General Public License, version 3 No. of lines in distributed program, including test data, etc.: 146353 No. of bytes in distributed program, including test data, etc.: 4890534 Distribution format: tar.gz Programming language: C++. Computer: Any computer that can run a C++ Compiler. Operating system: Tested on Microsoft Windows 7, Ubuntu 14.4. RAM: From a few Mb to several Gb depending on the size of the system being simulated. Classification: 22. External routines: Eigen, Boost C++ Libraries, RapidXML Nature of problem: Evolution of reacting gas mixtures with detailed description of thermodynamic, kinetic and transport data. Solution method: Stiff systems of Ordinary differential Equations, whose solution is obtained using methods based on the Backward Differentiation Formulas (BDF) (LU factorization of dense matrices is required). Additional comments: The code was specifically conceived for managing homogeneous, reacting mixtures including thousands of species and reactions. Running time: Problem-dependent, from seconds (small kinetics) to hours [ABSTRACT FROM AUTHOR]