This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: at +div(vf(u)) = 0 inIR￣N × [0, T], with initial data u(., 0) = ...This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: at +div(vf(u)) = 0 inIR￣N × [0, T], with initial data u(., 0) = uo(.) inIR￣N ) where uo ∈ L∞(IR￣N ) is a given function, v is a divergence-free bounded functioll of class C1 from IR￣× x [0, T] to IR￣N, and f is a function of class C1 from IR toIR. It also gives a result of convergence of a numerical scheme for the discretization of this equation. The authors first show the existence of a 'process' solution (which generalizes the concept of entropy weak solutions, and can be obtained by passing to the limit of solutions of the numerical scheme). The uniqueness of this entropy process solution is then proven; it is also proven that the entropy process solution is in fact an entropy weak solution. Hence the existence and uniqueness of the entropy weak solution are proven.展开更多
The approximation of problems with linear convection and degenerate nonlinear difFusion,which arise in the framework of the transport of energy in porous media with thermodynamic transitions,is done usingθ-scheme bas...The approximation of problems with linear convection and degenerate nonlinear difFusion,which arise in the framework of the transport of energy in porous media with thermodynamic transitions,is done usingθ-scheme based on the centred gradient discretisation method.The convergence of the numerical scheme is proved,although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field,owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting.Some numerical examples,using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme,show the role ofθfor stabilising the scheme.展开更多
文摘This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: at +div(vf(u)) = 0 inIR￣N × [0, T], with initial data u(., 0) = uo(.) inIR￣N ) where uo ∈ L∞(IR￣N ) is a given function, v is a divergence-free bounded functioll of class C1 from IR￣× x [0, T] to IR￣N, and f is a function of class C1 from IR toIR. It also gives a result of convergence of a numerical scheme for the discretization of this equation. The authors first show the existence of a 'process' solution (which generalizes the concept of entropy weak solutions, and can be obtained by passing to the limit of solutions of the numerical scheme). The uniqueness of this entropy process solution is then proven; it is also proven that the entropy process solution is in fact an entropy weak solution. Hence the existence and uniqueness of the entropy weak solution are proven.
基金supported by the French Agence Nationale de la Recherche(CHARMS project,ANR-16-CE06-0009).
文摘The approximation of problems with linear convection and degenerate nonlinear difFusion,which arise in the framework of the transport of energy in porous media with thermodynamic transitions,is done usingθ-scheme based on the centred gradient discretisation method.The convergence of the numerical scheme is proved,although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field,owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting.Some numerical examples,using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme,show the role ofθfor stabilising the scheme.
