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Structure-preserving shock-capturing methods: late-time asymptotics, curved geometry, small-scale dissipation, and nonconservative products

We present some recent developments on shock capturing methods for nonlinear hyperbolic systems of balance laws, whose prototype is the Euler system of compressible fluid flows, and especially discuss {structure-preserving} techniques. The problems under consideration arise with complex fluids in realistic applications when friction terms, geometrical terms, viscosity and capillarity effects, etc., need to be taken into account in order to achieve a proper description of the physical phenomena. For these problems, it is necessary to design numerical methods that are not only consistent with the given partial differential equations, but remain accurate and robust in certain {asymptotic regimes} of physical interest. That is, certain structural properties of these hyperbolic problems (conservation or balance law, equilibrium state, monotonicity properties, etc.) are essential in many applications, and one seeks that the numerical solutions preserve these properties, which is often a very challenging task.