Fast-wave slow-wave spectral deferred correction methods applied to the compressible Euler equations

Abstract

This paper investigates the application of a fast-wave slow-wave spectral deferred correction time-stepping method (FWSW-SDC) to the compressible Euler equations. The resulting model achieves arbitrary order accuracy in time, demonstrating robust performance in standard benchmark idealised test cases for dynamical cores used for numerical weather prediction. The model uses a compatible finite element spatial discretisation, achieving good linear wave dispersion properties without spurious computational modes. A convergence test confirms the model's high temporal accuracy. Arbitrarily high spatial-temporal convergence is demonstrated using a gravity wave test case. The model is further extended to include the parametrisation of a simple physics process by adding two phases of moisture and its validity is demonstrated for a rising thermal problem. Finally, a baroclinic wave in simulated in a Cartesian domain.

Figures (4)

• Figure 1: Temporal convergence of SDC for an advection test on the sphere. We use $\Delta t = [2400, 1800, 1200, 900]s$ on a C$32$ cubed sphere mesh. We used a self convergence test to eliminate the spatial error with a SSPRK3 scheme and $\Delta t_{true} = 0.5$s. $SDC(2,3)$ and $SDC(3,5)$ achieve the expected order of convergence of $4$, $6$ and $8$ respectively.
• Figure 2: (Right) Final $\theta^{'}$solution for the gravity-wave test case at $t=3000$s for $(\Delta x, \Delta z) =(2000, 1000)$m and $\Delta t = 6$s. Contours are spaced every $5 \times 10^{-4}$K. The solution is the same as in the literature melvin2010inherently, skamarock1994efficiency. (Left) Convergence of $\theta$$L_2$ error for the gravity-wave test case at $t=3000$s for a constant Courant–Friedrichs–Lewy (CFL) number of $0.06$ for the $p = 1$ case. The $p = 1$$SDC(2,3)$ scheme has second order convergence, $p = 3$$SDC(3,5)$ has better than fourth order convergence and $p = 5$$SDC(4,7)$ has better than fifth order convergence.
• Figure 3: $\theta_e$ field at time $t = 1000$ s for $Q^{\texttt{LU}}_{\Delta}$ and $Q^{\texttt{ExpEuler}}_{\Delta}$ (Left), and $Q^{\texttt{MIN-SR-FLEX}}_{\Delta}$ and $Q^{\texttt{MIN-SR-NS}}_{\Delta}$ (Right). As in bendall2020compatible, $\Delta x = \Delta z = 100$m and $\Delta t = 1$ s. The $320$K contour has been omitted, and the contours are spaced every $0.5$K.
• Figure 4: Temperature with contours from $268$K to $306$K with spacings of $2$K (top) and pressure with contours from $92600$Pa to $95800$Pa with spacings of $200$Pa (bottom) fields at $t=12$days $z=500$m, the resolutions were $\Delta t = 1800$s, $\Delta x = \Delta y = 250$km and $\Delta z = 1.5$km.