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This paper introduces AdHImEx, a novel Adaptively Implicit-Explicit (AdImEx) time stepping scheme that achieves second-order accuracy for large Courant numbers while requiring only a single sparse matrix solve per time step, overcoming limitations of prior first-order AdImEx methods. The scheme blends third-order explicit and Crank-Nicolson implicit Runge-Kutta stages, demonstrating verified stability for Courant numbers up to 100 and retaining explicit efficiency in low-Courant regions. Combined with a fifth-order finite-volume spatial discretization, AdHImEx offers substantial improvements in accuracy and efficiency for mass-conservative transport problems, particularly in atmospheric modeling.
Achieve second-order accuracy for mass transport with Courant numbers up to 100 using a single matrix solve per time step, outperforming prior AdImEx schemes.
Adaptively Implicit-Explicit (AdImEx) time stepping provides stability for large time steps for mass-conservative transport, making it attractive for the numerical representation of advection in weather and climate prediction. Existing AdImEx transport schemes become first-order accurate in the large-time-step, implicit regime and may require multiple sparse matrix solutions per time step. This work introduces AdHImEx, a new AdImEx scheme that addresses both limitations simultaneously. AdHImEx provides second-order accuracy for large Courant numbers while requiring only a single matrix solution per time step. It is a Runge-Kutta scheme that blends third-order accurate explicit time stepping and Crank-Nicolson implicit time stepping to provide numerically verified stability for Courant numbers up to 100. AdHImEx is designed for flows that are predominantly explicit, activating implicit time stepping only where large Courant numbers occur. It retains the full efficiency and third-order accuracy of the explicit time stepping in regions with small Courant numbers. In this work, it is combined with a fifth-order accurate finite-volume discretisation in space for mass conservation. A second contribution of this work is a novel stage-dependent treatment of the divergence operator that preserves constancy despite different implicit and explicit Runge-Kutta stage time steps and spatially varying implicitness, removing spurious divergence. This innovation is applicable to other AdImEx time stepping schemes. Convergence, transport, and efficiency tests demonstrate substantial improvements over previous first-order AdImEx schemes, little phase and amplitude error, accurate transport on locally refined meshes with large Courant numbers, and promising efficiency characteristics for atmospheric transport.