While quite different, many physical models can be described by nonlinear hyperbolic systems of conservation and balance laws. The main source of difficulties one comes across when numerically solving these systems is a lack of smoothness, as solutions of hyperbolic conservation/balance laws may develop very complicated nonlinear wave structures, including shocks, rarefaction waves, and contact discontinuities. The level of complexity may increase even further when solutions of the hyperbolic system reveal a multiscale character and/or the system includes additional terms such as friction terms, geometrical terms, nonconservative products, etc., which need to be taken into account to achieve a proper description of the studied physical phenomena. In such cases, it is essential to design a numerical method that is not only consistent with the given PDEs but also preserves specific structural and asymptotic properties of the underlying problem at the discrete level. While various numerical methods for such models have been successfully developed, there are still many open problems for which the derivation of reliable high-resolution numerical methods remains a highly challenging task.

In this talk, I will discuss recent advances in developing two classes of structure-preserving numerical methods for nonlinear hyperbolic systems of conservation and balance laws. In particular, I will present (i) well-balanced and positivity-preserving numerical schemes, that is, the methods that are capable of exactly preserving some steady-state solutions as well as maintaining the positivity of the numerical quantities when it is required by the physical application and (ii) asymptotic preserving schemes, which provide accurate and efficient numerical solutions in specific stiff and/or asymptotic regimes of physical interest.