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Weizhang Huang - One of the best experts on this subject based on the ideXlab platform.
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a quasi lagrangian moving mesh discontinuous galerkin method for hyperbolic Conservation Laws
Journal of Computational Physics, 2019Co-Authors: Weizhang HuangAbstract:Abstract A moving mesh discontinuous Galerkin method is presented for the numerical solution of hyperbolic Conservation Laws. The method is a combination of the discontinuous Galerkin method and the mesh movement strategy which is based on the moving mesh partial differential equation approach and moves the mesh continuously in time and orderly in space. It discretizes hyperbolic Conservation Laws on moving meshes in the quasi-Lagrangian fashion with which the mesh movement is treated continuously and no interpolation is needed for physical variables from the old mesh to the new one. Two convection terms are induced by the mesh movement and their discretization is incorporated naturally in the DG formulation. Numerical results for a selection of one- and two-dimensional scalar and system Conservation Laws are presented. It is shown that the moving mesh DG method achieves the second and third order of convergence for P 1 and P 2 elements, respectively, for problems with smooth solutions and is able to capture shocks and concentrate mesh points in non-smooth regions. Its advantage over uniform meshes and its insensitiveness to mesh smoothness are also demonstrated.
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a quasi lagrangian moving mesh discontinuous galerkin method for hyperbolic Conservation Laws
arXiv: Numerical Analysis, 2018Co-Authors: Weizhang HuangAbstract:A moving mesh discontinuous Galerkin method is presented for the numerical solution of hyperbolic Conservation Laws. The method is a combination of the discontinuous Galerkin method and the mesh movement strategy which is based on the moving mesh partial differential equation approach and moves the mesh continuously in time and orderly in space. It discretizes hyperbolic Conservation Laws on moving meshes in the quasi-Lagrangian fashion with which the mesh movement is treated continuously and no interpolation is needed for physical variables from the old mesh to the new one. Two convection terms are induced by the mesh movement and their discretization is incorporated naturally in the DG formulation. Numerical results for a selection of one- and two-dimensional scalar and system Conservation Laws are presented. It is shown that the moving mesh DG method achieves the theoretically predicted order of convergence for problems with smooth solutions and is able to capture shocks and concentrate mesh points in non-smooth regions. Its advantage over uniform meshes and its insensitiveness to mesh smoothness are also demonstrated.
Eitan Tadmor - One of the best experts on this subject based on the ideXlab platform.
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construction of approximate entropy measure valued solutions for hyperbolic systems of Conservation Laws
Foundations of Computational Mathematics, 2017Co-Authors: Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, Eitan TadmorAbstract:Entropy solutions have been widely accepted as the suitable solution framework for systems of Conservation Laws in several space dimensions. However, recent results in De Lellis and Szekelyhidi Jr (Ann Math 170(3):1417---1436, 2009) and Chiodaroli et al. (2013) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of Conservation Laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of Conservation Laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of Conservation Laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of Conservation Laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions.
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Construction of approximate entropy measure valued solutions for hyperbolic systems of Conservation Laws
Foundations of Computational Mathematics, 2015Co-Authors: Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, Eitan TadmorAbstract:Entropy solutions have been widely accepted as the suitable solution framework for systems of Conservation Laws in several space dimensions. However, recent results in \cite{CDL1,CDL2} have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that demonstrates that state of the art numerical schemes \emph{may not} necessarily converge to an entropy solution of systems of Conservation Laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of Conservation Laws, particularly in several space dimensions. Furthermore, we propose a more general notion, that of \emph{entropy measure valued solutions}, as an appropriate solution paradigm for systems of Conservation Laws. To this end, we present a detailed numerical procedure, which constructs stable approximations to entropy measure valued solutions and provide sufficient conditions that guarantee that these approximations converge to an entropy measure valued solution as the mesh is refined, thus providing a viable numerical framework for systems of Conservation Laws in several space dimensions. A large number of numerical experiments that illustrate the proposed schemes are presented and are utilized to examine several interesting properties of the computed entropy measure valued solutions.
Martin Z Bazant - One of the best experts on this subject based on the ideXlab platform.
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surface Conservation Laws at microscopically diffuse interfaces
Journal of Colloid and Interface Science, 2007Co-Authors: Martin Z BazantAbstract:In studies of interfaces with dynamic chemical composition, bulk and interfacial quantities are often coupled via surface Conservation Laws of excess surface quantities. While this approach is easily justified for microscopically sharp interfaces, its applicability in the context of microscopically diffuse interfaces is less theoretically well-established. Furthermore, surface Conservation Laws (and interfacial models in general) are often derived phenomenologically rather than systematically. In this article, we first provide a mathematically rigorous justification for surface Conservation Laws at diffuse interfaces based on an asymptotic analysis of transport processes in the boundary layer and derive general formulae for the surface and normal fluxes that appear in surface Conservation Laws. Next, we use nonequilibrium thermodynamics to formulate surface Conservation Laws in terms of chemical potentials and provide a method for systematically deriving the structure of the interfacial layer. Finally, we derive surface Conservation Laws for a few examples from diffusive and electrochemical transport.
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surface Conservation Laws at microscopically diffuse interfaces
arXiv: Chemical Physics, 2007Co-Authors: Martin Z BazantAbstract:In studies of interfaces with dynamic chemical composition, bulk and interfacial quantities coupled via surface Conservation Laws of excess surface quantities. While this approach is for microscopically sharp interfaces, its applicability in the context of microscopically diffuse is less theoretically well-established. Furthermore, surface Conservation Laws (and interfacial in general) are often derived phenomenologically rather than systematically. In this article, provide a mathematically rigorous justification for surface Conservation Laws at diffuse interfaces on an asymptotic analysis of transport processes in the boundary layer and derive general the surface and normal fluxes that appear in surface Conservation Laws. Next, we use non-thermodynamics to formulate surface Conservation Laws in terms of chemical potentials a method for systematically deriving the structure of the interfacial layer. Finally, we Conservation Laws for a few examples from diffusive and electrochemical transport.
Ulrik S. Fjordholm - One of the best experts on this subject based on the ideXlab platform.
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construction of approximate entropy measure valued solutions for hyperbolic systems of Conservation Laws
Foundations of Computational Mathematics, 2017Co-Authors: Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, Eitan TadmorAbstract:Entropy solutions have been widely accepted as the suitable solution framework for systems of Conservation Laws in several space dimensions. However, recent results in De Lellis and Szekelyhidi Jr (Ann Math 170(3):1417---1436, 2009) and Chiodaroli et al. (2013) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of Conservation Laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of Conservation Laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of Conservation Laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of Conservation Laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions.
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Construction of approximate entropy measure valued solutions for hyperbolic systems of Conservation Laws
Foundations of Computational Mathematics, 2015Co-Authors: Ulrik S. Fjordholm, Roger Käppeli, Siddhartha Mishra, Eitan TadmorAbstract:Entropy solutions have been widely accepted as the suitable solution framework for systems of Conservation Laws in several space dimensions. However, recent results in \cite{CDL1,CDL2} have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that demonstrates that state of the art numerical schemes \emph{may not} necessarily converge to an entropy solution of systems of Conservation Laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of Conservation Laws, particularly in several space dimensions. Furthermore, we propose a more general notion, that of \emph{entropy measure valued solutions}, as an appropriate solution paradigm for systems of Conservation Laws. To this end, we present a detailed numerical procedure, which constructs stable approximations to entropy measure valued solutions and provide sufficient conditions that guarantee that these approximations converge to an entropy measure valued solution as the mesh is refined, thus providing a viable numerical framework for systems of Conservation Laws in several space dimensions. A large number of numerical experiments that illustrate the proposed schemes are presented and are utilized to examine several interesting properties of the computed entropy measure valued solutions.
Nail H. Ibragimov - One of the best experts on this subject based on the ideXlab platform.
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nonlinear self adjointness Conservation Laws and exact solutions of time fractional kompaneets equations
Communications in Nonlinear Science and Numerical Simulation, 2015Co-Authors: R K Gazizov, Nail H. Ibragimov, Yu S LukashchukAbstract:Four time-fractional generalizations of the Kompaneets equation are considered. Group analysis is performed for physically relevant approximations. It is shown that all approximations have nontrivial symmetries and Conservation Laws. The symmetries are used for constructing group invariant solutions, whereas the Conservation Laws allow to find non-invariant exact solutions.
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Nonlinear self-adjointness in constructing Conservation Laws
arXiv: Mathematical Physics, 2011Co-Authors: Nail H. IbragimovAbstract:The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the previous notions of self-adjoint and quasi self-adjoint nonlinear equations. The class of nonlinearly self-adjoint equations includes, in particular, all linear equations. Conservation Laws associated with symmetries can be constructed for all nonlinearly self-adjoint differential equations and systems. The number of equations in systems can be different from the number of dependent variables.
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Lagrangian approach to evolution equations: symmetries and Conservation Laws.
Nonlinear Dynamics, 2004Co-Authors: Nail H. Ibragimov, Torbjörn KolsrudAbstract:We show that one can apply a Lagrangian approach to certain evolution equations by considering them together with their associated equations. Consequently, one can employ Noether's theorem and derive Conservation Laws from symmetries of coupled systems of evolution equations. We discuss in detail the linear and non-linear heat equations as well as the Burgers equation and obtain new non-local Conservation Laws for the non-linear heat and the Burgers equations by extending their symmetries to the associated equations. We also provide Lagrangians for non-linear Schrodinger and Korteweg—de Vries type systems.
