References

O. Guba, M. Taylor and A. St-Cyr. Optimization-based limiters for the spectral element method. Journal of computational physics 267, 176–195 (2014). ↩¹ ↩²

I. Higueras and T. Roldán. New third order low-storage SSP explicit Runge-Kutta methods, arXiv preprint arXiv:1809.04807 (2018). ↩¹

U. M. Ascher, S. J. Ruuth and R. J. Spiteri. Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics 25, 151–167 (1997). ↩¹ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴ ↩¹⁵ ↩¹⁶ ↩¹⁷

F. X. Giraldo, J. F. Kelly and E. M. Constantinescu. Implicit-explicit formulations of a three-dimensional nonhydrostatic unified model of the atmosphere (NUMA). SIAM Journal on Scientific Computing 35, B1162–B1194 (2013). ↩¹ ↩² ↩³ ↩⁴

C. A. Kennedy and M. H. Carpenter. Higher-order additive Runge–Kutta schemes for ordinary differential equations. Applied Numerical Mathematics 136, 183–205 (2019). ↩¹ ↩² ↩³ ↩⁴

C. J. Vogl, A. Steyer, D. R. Reynolds, P. A. Ullrich and C. S. Woodward. Evaluation of implicit-explicit additive runge-kutta integrators for the HOMME-NH dynamical core. Journal of Advances in Modeling Earth Systems 11, 4228–4244 (2019). ↩¹ ↩²

O. Guba, M. A. Taylor, A. M. Bradley and A. Steyer. A framework to evaluate IMEX schemes for atmospheric models. Geoscientific Model Development 13, 6467–6480 (2020). ↩¹ ↩²

A. J. Steyer, C. J. Vogl, M. Taylor and O. Guba. Efficient IMEX Runge-Kutta methods for nonhydrostatic dynamics, arXiv (2019). ↩¹ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹ ↩¹⁰ ↩¹¹ ↩¹² ↩¹³ ↩¹⁴ ↩¹⁵ ↩¹⁶ ↩¹⁷ ↩¹⁸ ↩¹⁹ ↩²⁰ ↩²¹ ↩²² ↩²³ ↩²⁴ ↩²⁵ ↩²⁶ ↩²⁷ ↩²⁸ ↩²⁹

L. Pareschi and G. Russo. Implicit–explicit Runge–Kutta schemes and applications to hyperbolic systems with relaxation. Journal of Scientific Computing 25, 129–155 (2005). ↩¹ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶

S. Conde, S. Gottlieb, Z. J. Grant and J. N. Shadid. Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order. Journal of Scientific Computing 73, 667–690 (2017). ↩¹ ↩²

E. Hairer and G. Wanner. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems. 2nd Edition (Springer-Verlag, Berlin, Heidelberg, 1996). ↩¹ ↩² ↩³

J. H. Williamson. Low-storage Runge-Kutta schemes. Journal of Computational Physics 35, 48–56 (1980). ↩¹

M. H. Carpenter and C. A. Kennedy. Fourth-order $2N$-storage Runge–Kutta schemes. Technical Report NASA TM-109112 (National Aeronautics and Space Administration, Langley Research Center, Hampton, VA, 1994). ↩¹ ↩² ↩³

J. Niegemann, R. Diehl and K. Busch. Efficient low-storage Runge–Kutta schemes with optimized stability regions. Journal of Computational Physics 231, 364–372 (2012). ↩¹ ↩² ↩³

M. Schlegel, O. Knoth, M. Arnold and R. Wolke. Implementation of multirate time integration methods for air pollution modelling. Geoscientific Model Development 5, 1395–1405 (2012). ↩¹

J. Wensch, O. Knoth and A. Galant. Multirate infinitesimal step methods for atmospheric flow simulation. BIT Numerical Mathematics 49, 449–473 (2009). ↩¹ ↩²

O. Knoth and J. Wensch. Generalized split-explicit Runge–Kutta methods for the compressible Euler equations. Monthly Weather Review 142, 2067–2081 (2014). ↩¹ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸ ↩⁹

L. J. Wicker and W. C. Skamarock. A time-splitting scheme for the elastic equations incorporating second-order Runge–Kutta time differencing. Monthly Weather Review 126, 1992–1999 (1998). ↩¹ ↩² ↩³ ↩⁴

L. J. Wicker and W. C. Skamarock. Time-splitting methods for elastic models using forward time schemes. Monthly Weather Review 130, 2088–2097 (2002). ↩¹ ↩² ↩³ ↩⁴

D. J. Gardner, J. E. Guerra, F. P. Hamon, D. R. Reynolds, P. A. Ullrich and C. S. Woodward. Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models. Geoscientific Model Development 11, 1497–1515 (2018). ↩¹ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷ ↩⁸

C.-W. Shu and S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of computational physics 77, 439–471 (1988). ↩¹ ↩²

K. Heun. Neue Methoden zur approximativen Integration der Differentialgleichungen einer unabhängigen Veränderlichen. Z. Math. Phys 45, 23–38 (1900). ↩¹

E. Süli and D. Mayers. An Introduction to Numerical Analysis (Cambridge University Press, 2003); p. 352. ↩¹

P. H. Lauritzen, W. C. Skamarock, M. J. Prather and M. A. Taylor. A standard test case suite for two-dimensional linear transport on the sphere. Geoscientific Model Development 5, 887–901 (2012). ↩¹