Talk by Arpit Babbar
Posted on: 20 Dec 2025
Title: Compact Runge-Kutta methods for non-conservative hyperbolic equations
Speaker: Arpit Babbar, Humboldt Postdoc, Johannes Gutenberg University, Mainz
Date: 5 Jan 2026
Time: 2 PM IST
Venue: Auditorium, TIFR-CAM and on Zoom
Abstract: Compact Runge–Kutta (cRK) methods are space–time discretizations that use stencils involving only immediate neighboring elements. Application of cRK methods has so far been limited to conservative equations. In this work, we construct a quadrature-free cRK scheme using the Flux Reconstruction (FR) framework, referred to as the cRKFR scheme, for the numerical solution of general hyperbolic equations with stiff source terms and non-conservative products.
To treat stiff source terms, we use IMplicit–EXplicit (IMEX) time integration schemes in which the implicit solves are local to each solution point, thereby avoiding additional inter-element communication. Non-conservative products are handled through a formulation based on discontinuous numerical fluxes at element interfaces. The same flux formulation is used to define a lower-order finite volume scheme on subcells, which is blended with the high-order cRKFR scheme to approximate non-smooth solutions. When combined with a flux limiter, the subcell based blending scheme preserves physical admissibility, such as positivity of density and pressure in the compressible Euler equations.
The proposed approach results in an admissibility-preserving IMEX cRKFR scheme. Its ability to handle stiff source terms is demonstrated through numerical experiments involving Burgers’ equation, the reactive Euler equations, and the ten-moment model. The treatment of non-conservative products is validated using variable-coefficient advection, shear shallow water equations, GLM magnetohydrodynamics, and multi-ion MHD equations.