Attempts to define weak solutions to nonconservative hyperbolic systems have lead to the development of several approaches, most notably the path-based theory of Dal Maso, LeFloch, and Murat (DLM) and the vanishing viscosity solutions described by Bianchini and Bressan. While these theories enable us to define weak solutions to nonconservative hyperbolic systems, difficulties arise when numerically approximating these systems. Specifically, in the neighborhood of a discontinuity, the numerical solutions tend to not converge to the theoretically specified weak solution of the system. This convergence error is easily seen in the numerical approximation of Riemann problems, in which the error appears and propagates at the formation of discontinuity waves. In this paper we investigate several methods to numerically approximate nonconservative hyperbolic systems, we discuss why these convergence errors arise, and by using recent results established by Alouges and Merlet we give an approximate description of what weak solutions these numerical solutions converge to. We then propose several strategies for the design of numerical schemes which reduce these convergence errors.

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Keywords Finite volume schemes, Godunov scheme, Hyperbolic systems, Lax-Friedrichs scheme, Nonconservative product
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Journal Journal of Computational Science
Chalmers, N., & Lorin, E. (2013). On the numerical approximation of one-dimensional nonconservative hyperbolic systems. Journal of Computational Science, 4(1-2), 111–124. doi:10.1016/j.jocs.2012.08.002