Semi-Implicit Formulations of the Navier–Stokes Equations: Application to Nonhydrostatic Atmospheric Modeling

Abstract

Abstract. We present semi-implicit (IMEX) formulations of the compressible Navier-Stokes equations (NSE) for applications in nonhydrostatic atmospheric modeling. The compressible NSE in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the Schur complement form of the linear implicit problem. We present results for five different forms of the compressible NSE and describe in detail how to formulate the semi-implicit time-integration method for these equations. Finally, we compare all five equations and compare the semi-implicit formulations of these equations both using the Schur and No Schur forms against an explicit Runge-Kutta method. Our simulations show that, if efficiency is the main criterion, it matters which form of the governing equations you choose. Furthermore, the semi-implicit formulations are faster than the explicit RungeKutta method for all the tests studied especially if the Schur form is used. While we have used the spectral element method for discretizing the spatial operators, the semi-implicit formulations that we derive are directly applicable to all other numerical methods. We show results for our five semi-implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling, including: inertia gravity waves, rising thermal bubbles (i.e., Rayleigh-Taylor instabilities), density current (i.e., Kelvin-Helmholtz instabilities), and mountain test cases; the latter test case requires the implementation of non-reflecting boundary conditions. Therefore, we show results for all five semi-implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling: no-flux (reflecting) and non-reflecting boundary conditions. It is shown that the non-reflecting boundary conditions exert a strong impact on the accuracy and efficiency of the models.Key words. compressible flow; element-based Galerkin methods; Euler; IMEX; Lagrange; Legendre; NavierStokes; nonhydrostatic; spectral elements; time-integration.AMS subject classifications. 65M60, 65M70, 35L65, 86A101. Introduction. It can be argued that the single most important property of an operational nonhydrostatic mesoscale atmospheric is efficiency. Clearly, this efficiency should not come at the cost of accuracy but if a weather center has the choice between a very accurate model and one that is efficient, they will probably pick the efficient one; however, as numerical analysts, we would like to build models that are both accurate and efficient. One way to achieve this goal is to construct numerical models based on high-order methods: this class of methods offers exponential (spectral) convergence for smooth problems and achieves excellent scalability on modern multi-core systems if they are used in an element-based approach (i.e., if the approximating polynomials have compact/local support). This is the idea behind element-based Galerkin methods such as spectral element (SE) and discontinuous Galerkin (DG) methods (see [14] and [2...

Implicit–Explicit Multistep Methods for Fast-Wave–Slow-Wave Problems

“…The condition number is influenced by both the implicit part of the IMEX scheme and by the time step. Giraldo (2005) noted that, provided the condition number is small enough to dramatically reduce the number of iterations required during each implicit step, the most efficient scheme can sometimes be one for which n 1 /a 1 and the maximum time step are both small. The values of (n 1 /a 1 ) 2 for the preceding implicit methods are listed in Table 5.…”

“…Choosing b 5 0 yields a method that has seen use in combination with BDF2 to obtain second-order IMEX methods (Karniadakis et al 1991;Giraldo et al 2010). …”

“…The choice b 5 2 /3 gives a thirdorder method that will be denoted BX3*; this method is the explicit component of the IMEX BDF2B scheme of Giraldo (2005), who attributed it to Hulsten. Figure 3 shows jAj for selected explicit methods in the same format as Fig.…”

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

“…Among the five equation sets discussed in [16], NUMA uses three: two conservative sets (Set2C and Set3C) used with both CG and DG discretizations, and a non-conservative equation set used exclusively with CG (Set2NC). In the following, we give an outline of equation Set2C.…”

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

“…The system of Eq. (1) is just one of several possibilities to write the equations governing the atmosphere (for alternatives see, for instance, [41,42]). In addition, here we do not formally define the physical parametrization terms P and Q, as they are not relevant for the purpose of this review.…”