Positivity-preserving numerical scheme for hyperbolic systems with $$\delta $$ -shock solutions and its convergence analysis
Tóm tắt
In this article, numerical schemes are proposed for approximating the solutions, possibly measure-valued with concentration (delta shocks), for a class of nonstrictly hyperbolic systems. These systems are known to model physical phenomena such as the collision of clouds and dynamics of sticky particles, for example. The scheme is constructed by extending the theory of discontinuous flux for scalar conservation laws, to capture measure-valued solutions with concentration. The numerical approximations are analytically shown to be entropy stable in the framework of Bouchut (Adv Kinet Theory Comput 22:171–190, 1994), satisfy the physical properties of the state variables, and converge to the weak solution. The construction allows natural extensions of the scheme to its higher-order and multi-dimensional versions. The scheme is also extended for some more classes of fluxes, which admit delta shocks and are also known to model physical phenomena. Various physical systems are simulated both in one dimension and multi-dimensions to display the performance of the numerical scheme, and comparisons are made with the test problems available in the literature.
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