Search results for "Conservation laws"
showing 6 items of 6 documents
Nonlinear Diffusion in Transparent Media
2021
Abstract We consider a prototypical nonlinear parabolic equation whose flux has three distinguished features: it is nonlinear with respect to both the unknown and its gradient, it is homogeneous, and it depends only on the direction of the gradient. For such equation, we obtain existence and uniqueness of entropy solutions to the Dirichlet problem, the homogeneous Neumann problem, and the Cauchy problem. Qualitative properties of solutions, such as finite speed of propagation and the occurrence of waiting-time phenomena, with sharp bounds, are shown. We also discuss the formation of jump discontinuities both at the boundary of the solutions’ support and in the bulk.
A numerical study of postshock oscillations in slowly moving shock waves
2003
Abstract Godunov-type methods and other shock capturing schemes can display pathological behavior in certain flow situations. This paper discusses the numerical anomaly associated to slowly moving shocks. We present a series of numerical experiments that illustrate the formation and propagation of this pathology, and allows us to establish some conclusions and question some previous conjectures for the source of the numerical noise. A simple diagnosis on an explicit Steger-Warming scheme shows that some intermediate states in the first time steps deviate from the true direction and contaminate the flow structure. A remedy is presented in the form of a new flux split method with an entropy i…
Lagrangian dynamics and possible isochronous behavior in several classes of non-linear second order oscillators via the use of Jacobi last multiplier
2015
Abstract In this paper, we employ the technique of Jacobi Last Multiplier (JLM) to derive Lagrangians for several important and topical classes of non-linear second-order oscillators, including systems with variable and parametric dissipation, a generalized anharmonic oscillator, and a generalized Lane–Emden equation. For several of these systems, it is very difficult to obtain the Lagrangians directly, i.e., by solving the inverse problem of matching the Euler–Lagrange equations to the actual oscillator equation. In order to facilitate the derivation of exact solutions, and also investigate possible isochronous behavior in the analyzed systems, we next invoke some recent theoretical result…
Generalized Camassa-Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions
2021
In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the…
AN HYPERBOLIC-PARABOLIC PREDATOR-PREY MODEL INVOLVING A VOLE POPULATION STRUCTURED IN AGE
2020
Abstract We prove existence and stability of entropy solutions for a predator-prey system consisting of an hyperbolic equation for predators and a parabolic-hyperbolic equation for preys. The preys' equation, which represents the evolution of a population of voles as in [2] , depends on time, t, age, a, and on a 2-dimensional space variable x, and it is supplemented by a nonlocal boundary condition at a = 0 . The drift term in the predators' equation depends nonlocally on the density of preys and the two equations are also coupled via classical source terms of Lotka-Volterra type, as in [4] . We establish existence of solutions by applying the vanishing viscosity method, and we prove stabil…
An order-adaptive compact approximation Taylor method for systems of conservation laws
2021
Abstract We present a new family of high-order shock-capturing finite difference numerical methods for systems of conservation laws. These methods, called Adaptive Compact Approximation Taylor (ACAT) schemes, use centered ( 2 p + 1 ) -point stencils, where p may take values in { 1 , 2 , … , P } according to a new family of smoothness indicators in the stencils. The methods are based on a combination of a robust first order scheme and the Compact Approximate Taylor (CAT) methods of order 2p-order, p = 1 , 2 , … , P so that they are first order accurate near discontinuities and have order 2p in smooth regions, where ( 2 p + 1 ) is the size of the biggest stencil in which large gradients are n…