Schemes with Well-Controlled Dissipation. Hyperbolic Systems in Nonconservative Form (Communications in Computational Physics, 2018)
We propose here a class of numerical schemes for the approximation ofweak solutions to nonlinear hyperbolic systems in nonconservative form – the notionof solution being understood in the sense of Dal Maso, LeFloch, and Murat (DLM).The proposed numerical method falls within LeFloch-Mishra's framework of schemeswith well-controlled dissipation (WCD), recently introduced for dealing with small-scaledependent shocks. We design WCD schemes which are consistent with a givennonconservative system at arbitrarily high-order and then analyze their linear stability.We then investigate several nonconservative hyperbolic models arising in complexfluid dynamics, and we numerically demonstrate the convergence of our schemes towardphysically meaningful weak solutions.