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Register a new userQuasi-static limit for a hyperbolic conservation law
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We study the quasi-static limit for the $L^infty$ entropy weak solution of scalar one-dimensional hyperbolic equations with strictly concave or convex flux and time dependent boundary conditions. The quasi-stationary profile evolves with the quasi-static equation, whose entropy solution is determined by the stationary profile corresponding to the boundary data at a given time.
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We consider a general conservation law on the circle, in the presence of a sublinear damping. If the damping acts on the whole circle, then the solution becomes identically zero in finite time, following the same mechanism as the corresponding ordinary differential equation. When the damping acts only locally in space, we show a dichotomy: if the flux function is not zero at the origin, then the transport mechanism causes the extinction of the solution in finite time, as in the first case. On the other hand, if zero is a non-degenerate critical point of the flux function, then the solution becomes extinct in finite time only inside the damping zone, decays algebraically uniformly in space, and we exhibit a boundary layer, shrinking with time, around the damping zone. Numerical illustrations show how similar phenomena may be expected for other equations.
We show the convergence of the zero relaxation limit in systems of $2 times 2$ hyperbolic conservation laws with stochastic initial data. Precisely, solutions converge to a solution of the local equilibrium approximation as the relaxation time tends to zero. The initial data are assumed to depend on finitely many random variables, and the convergence is then proved via the appropriate analogues of the compensated compactness methods used in treating the deterministic case. We also demonstrate the validity of this limit in the case of the semi-linear $p$-system; the well-posedness of both the system and its equilibrium approximation are proved, and the convergence is shown with no a priori conditions on solutions. This model serves as a prototype for understanding how asymptotic approximations can be used as control variates for hyperbolic balance laws with uncertainty.
We prove the decay of the L 2-distance from the solution u(t) of a hyperbolic scalar conservation law, to some convex, flow-invariant target sets.
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.
We introduce a formulation of the initial and boundary value problem for nonlinear hyperbolic conservation laws posed on a differential manifold endowed with a volume form, possibly with a boundary; in particular, this includes the important case of Lorentzian manifolds. Only limited regularity is assumed on the geometry of the manifold. For this problem, we establish the existence and uniqueness of an L1 semi-group of weak solutions satisfying suitable entropy and boundary conditions.
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