Low-storage, explicit Runge–Kutta schemes for the compressible Navier–Stokes equations

Abstract

variables needed to evaluate the flux terms. Consequently, the U-vector nmst be decomposed into other variables, leaving the (:-vector itself disposable. Both Williamson (2N) and vdH (2R) schemes may be easily generalized I() accommodate more than two storage registers (N or R). We make no claim that these two strategies are the only viable ones. We do suggest, however, that the vdH methodology is extremely aggressive in its conservation of computer memory usage. In t.he pursui! of computer memory use reduction, the first, casualty is the retention of the U-vector at. the beginning stage. Error control, in the more traditional sense, becomes impossible. A rejected step (such as violation of the error tolerance) cannot be restarted front U (') because with a 2N or 2t1 scheme, U (') is no tongrr available. Instead, alloting one additional register for an error estimate, one may monitor the error nlathematical soft.ware.J18, 28, 29, 58, 88] Attempts were made to solve for schemes symbolically, but il was found that the assumption of various a O = bj quickly made matters intractable because the equations of condition become algebraically nonlinear in the bi's. Scheme coefficients are given to at. lea,st 2,5 digils of accuracy. Some Ell.I{ background is necessary to facilitate a discussion on the optimization of accuracy efficiency, stability efficiency, error control reliability, step-control stability, linear stability, nonlinear stability, and dispersion and dissipation error within the context, of storage reduction in later chapters. This will be done in sections 1 and 2. Two-register schemes will be reviewed in section 3 while three-, four-, and five-register schemes will be considered in sections 4, .5, and 6. Merits of the low-storage schemes are discussed in section P (mininmm phase error), and for "P" melhods, qdisp and qdis_ are the respective dispersion and dissipation orders of accuracy.

Supergranulation Scale Convection Simulations

“…Spatial derivatives were evaluated using sixth order finite differences [1] and the time advance was by a low memory, third order Runge-Kutta scheme [2]. The calculations were performed on a grid of 1000 2 ¤ 500 giving a resolution of 96 km horizontally and 12-70 km vertically.…”

On high order strong stability preserving runge-kutta and multi step time discretizations

“…Therefore low storage RK methods ( [1,15,37]), which only require 2 or 3 storage units per ODE variable, may be desirable. In [7,8,26,27], some SSP low storage RK methods were studied.…”

“…Some of these methods are guaranteed optimal, others are the best found in extensive numerical searches. Ruuth considered Williamson schemes [37] which require two units of storage per step and Van Der Houwen and Wray (described in [15]) schemes requiring two or three registers of storage. Negative β i,k s are allowed in these methods, as long as all the negative coefficients are associated with the same superscript, so that for each U k either L orL appears, but not both, and so the low storage property is not destroyed.…”

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

“…Both subcategories, multistage Runge-Kutta and linear multistep methods, can be fullyimplicit (the current time-level is obtained by solving a nonlinear problem that uses information from the current time-step) and fully-explicit (the current time-level is calculated using information coming from the previous time-steps only). Representative classes of time-integration schemes embedded in the GL method consist of implicit multistep methods such as Adams-Moulton (AM) [22] and backward differentiation (BDF) methods [13,20,21], implicit multistage Runge-Kutta schemes such as diagonally (DIRK) and singly-diagonally (SDIRK) implicit Runge-Kutta schemes [3,19,59], explicit multistep methods, such as leapfrog and Adams-Bashforth methods [28,43], explicit Runge-Kutta schemes, such as the fourthorder Runge-Kutta scheme [55] and partitioned methods, such as Implicit-Explicit (IMEX) schemes, whereby the operators are linearized in some fashion with-e.g., two Butcher tableaux, one explicit and one implicit [5,40,106]. While EBTI schemes are widely used in computational fluid dynamics, especially in the engineering sector [18,52], their adoption in the weather and climate communities has been less widespread, with SE schemes [54,88,107] and horizontally-explicit vertically-implicit schemes [8,40,63]-i.e., schemes where the horizontal direction is treated explicitly and the vertical is treated implicitly-becoming more prominent but still confined mainly to research and limited-area models (with very few exceptions-see Table 1).…”