Second-order Optimally Stable IMEX (pseudo-)staggered Galerkin discretization: application to lava flow modeling

Abstract: We present second-order optimally stable Implicit-Explicit (IMEX) Runge-Kutta (RK) schemes with application to a modified set of shallow water equations that can be used to model the dynamics of lava flows. The schemes are optimally stable in the sense that they satisfy, at the space-time discretization level, a condition analogous to the \texttt{L}-stability of Runge-Kutta methods for ordinary differential equations. A novel (pseudo-)staggered Galerkin scheme is introduced, which can be interpreted as an extension of the classical two-step Taylor-Galerkin (TG2) scheme. The method is derived by combining a von Neumann stability analysis with a Lax-Wendroff procedure. For the discretization of the non-conservative terms that characterize the lava flow model, we employ the Path-Conservative (PC) method. The proposed scheme is evaluated on a number of relevant test cases, demonstrating accuracy, robustness, and well-balancing properties for the lava flow model.