Asymptotic-Preserving and Well-Balanced Linearly Implicit IMEX Schemes for the Anelastic Limit of the Isentropic Euler Equations with Gravity
This work provides a novel numerical method for simulating low-Mach-number atmospheric flows, addressing the need for efficient and accurate schemes in geophysical fluid dynamics.
The authors develop higher-order linearly implicit IMEX Runge-Kutta schemes for the compressible Euler system with anelastic scaling and gravity, achieving asymptotic preservation and well-balancing. Numerical results validate the theoretical properties.
We consider the compressible Euler system with anelastic scaling, modeling isentropic flows under the influence of gravity. In the zero-Mach-number limit, the solution of the compressible Euler system converges to a variable density anelastic incompressible limit system. In this work, we present the design and analysis of a class of higher-order linearly implicit IMEX Runge-Kutta schemes that are asymptotic preserving, i.e., they respect the transitory nature of the governing equations in the limit. The presence of gravitational potential warrants the incorporation of the well-balancing property. The scheme is developed as a novel combination of a penalization of a linear steady state, a finite-volume balance-preserving reconstruction, and a source term discretization preserving steady states. The penalization plays a crucial role in obtaining a linearly implicit scheme, and well-balanced flux-source discretization ensures accuracy in very low Mach number regimes. Some results of numerical case studies are presented to corroborate the theoretical assertions.