ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows

We develop a new family of well-balanced path-conservative quadrature-free one-step ADER finite volume and discontinuous Galerkin finite element schemes on unstructured meshes for the solution of hyperbolic partial differential equations with non-conservative products and stiff source terms. The fully discrete formulation is derived using the recently developed framework of explicit one-step PNPM schemes of arbitrary high order of accuracy in space and time for conservative hyperbolic systems [Dumbser M, Balsara D, Toro EF, Munz CD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:8209-53]. The two key ingredients of our high order approach are: first, the high order accurate PNPM reconstruction operator on unstructured meshes, using the WENO strategy presented in [Dumbser M, Kaser M, Titarev VA Toro EF. Quadrature-free non-oscillatory finite volume schemes on unstructured meshes for nonlinear hyperbolic systems. J Comput Phys 2007;226:204-43] to ensure monotonicity at discontinuities, and second, a local space-time Galerkin scheme to predict the evolution of the reconstructed polynomial data inside each element during one time step to obtain a high order accurate one-step time discretization. This approach is also able to deal with stiff source terms as shown in [Dumbser M, Enaux C, Toro EF. Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J Comput Phys 2008;227:3971-4001]. These two key ingredients are combined with the recently developed path-conservative methods of Pares [Pares C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework. SIAM J Numer Anal 2006;44:300-21] and Castro et al. [Castro MJ, Gallardo JM, Pares C. High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems. Math Comput 2006:75:1103341 to treat the non-conservative products properly. We show applications of our method to the two-layer shallow water equations as well as applications to the recently published depth-averaged two-fluid flow model of Pitman and Le I Pitman EB. Le L. A two-fluid model for avalanche and debris flows. Philos Trans Roy Soc A 2005:363:1573-601]. (C) 2009 Elsevier Ltd. All rights reserved.