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the efficiency with the addition of more processors. Table 2 Times per single iteration in seconds derived in a single observation test of the GSI using recursive filtering (RF) and the multigrid beta filter (MF) running with different numbers of processors (PEs). 6. Conclusions This paper summarizes work on development of a new method for modeling of background error covariance for application in the 3D RTMA system as a replacement for the recursive filter. The key prerequisite
the efficiency with the addition of more processors. Table 2 Times per single iteration in seconds derived in a single observation test of the GSI using recursive filtering (RF) and the multigrid beta filter (MF) running with different numbers of processors (PEs). 6. Conclusions This paper summarizes work on development of a new method for modeling of background error covariance for application in the 3D RTMA system as a replacement for the recursive filter. The key prerequisite
-grid models) but that this can be achieved using a single model driver, thus greatly simplifying operational requirements. An additional advantage of a two-way nested paradigm is that this can be the framework for exchanging information across different types of grids (regular, curvilinear, unstructured, etc.), thus having a modeling system that provides flexibility in choosing grids that best fit the local domains (e.g., unstructured grids in coastal waters). Apart from the multigrid approach, the third
-grid models) but that this can be achieved using a single model driver, thus greatly simplifying operational requirements. An additional advantage of a two-way nested paradigm is that this can be the framework for exchanging information across different types of grids (regular, curvilinear, unstructured, etc.), thus having a modeling system that provides flexibility in choosing grids that best fit the local domains (e.g., unstructured grids in coastal waters). Apart from the multigrid approach, the third
nested-grid models, multigrid methods ( Brandt 1977 ) use multiple grids of different mesh size covering the same region. The goal here is to solve the discretized equations with optimum efficiency: using simple relaxation on each grid to smooth the error on the scale of that grid produces the solution (on the finest grid) to the level of truncation error in computational work proportional to the number of unknowns (usually in the work of just a few relaxation sweeps). In addition to speeding up
nested-grid models, multigrid methods ( Brandt 1977 ) use multiple grids of different mesh size covering the same region. The goal here is to solve the discretized equations with optimum efficiency: using simple relaxation on each grid to smooth the error on the scale of that grid produces the solution (on the finest grid) to the level of truncation error in computational work proportional to the number of unknowns (usually in the work of just a few relaxation sweeps). In addition to speeding up
portable operational alternative to VICBAR. Some characteristics of these models are given in Table 1 . Recently a new multigrid barotropic model (MUDBAR) has been developed ( Fulton 2001 ). Based on the simple modified barotropic vorticity equation, MUDBAR uses an adaptive multigrid method to refine the mesh around the moving vortex, with the goal of maximizing accuracy while minimizing computational cost. MUDBAR can also be run in a mode that uses fixed-size movable meshes. This model was developed
portable operational alternative to VICBAR. Some characteristics of these models are given in Table 1 . Recently a new multigrid barotropic model (MUDBAR) has been developed ( Fulton 2001 ). Based on the simple modified barotropic vorticity equation, MUDBAR uses an adaptive multigrid method to refine the mesh around the moving vortex, with the goal of maximizing accuracy while minimizing computational cost. MUDBAR can also be run in a mode that uses fixed-size movable meshes. This model was developed
and Brown 2002 ; Brown 2004 ). The mathematical formulation in the mass-consistent model is a Poisson equation for the Lagrange multiplier as described below. All models reviewed above employ the traditional Gauss–Seidel (GS) iteration method. The convergence of the method is too slow for real-time simulations when a domain has a large number of grid points. In recent years, the multigrid method has been used widely in solving the Poisson equation. There are several other popular iterative
and Brown 2002 ; Brown 2004 ). The mathematical formulation in the mass-consistent model is a Poisson equation for the Lagrange multiplier as described below. All models reviewed above employ the traditional Gauss–Seidel (GS) iteration method. The convergence of the method is too slow for real-time simulations when a domain has a large number of grid points. In recent years, the multigrid method has been used widely in solving the Poisson equation. There are several other popular iterative
correction, and postrelaxation. Figure 2 shows the procedure of a three-level V( N 1 , N 2 )-cycle multigrid method, where N 1 and N 2 are the numbers of relaxation sweeps for prerelaxation and postrelaxation, respectively, and a brief description is given in appendix A . As mentioned earlier, the implicit scheme is applied to both acoustic and gravity waves, and therefore a set of linear equations with a huge multiband sparse matrix needs to be solved. Modelers have been looking for a method
correction, and postrelaxation. Figure 2 shows the procedure of a three-level V( N 1 , N 2 )-cycle multigrid method, where N 1 and N 2 are the numbers of relaxation sweeps for prerelaxation and postrelaxation, respectively, and a brief description is given in appendix A . As mentioned earlier, the implicit scheme is applied to both acoustic and gravity waves, and therefore a set of linear equations with a huge multiband sparse matrix needs to be solved. Modelers have been looking for a method
solver. Even more recently, it has been shown that Helmholtz problems and Poisson problems of the sort arising in atmospheric modeling can be solved efficiently and with good parallel scalability using geometric multigrid methods ( Heikes et al. 2013 ; Müller and Scheichl 2014 ; Dedner et al. 2015, manuscript submitted to Int. J. Numer. Methods Fluids ). See, for example, Fulton et al. (1986) for a clear introduction to multigrid methods in the context of atmospheric modeling. Such methods
solver. Even more recently, it has been shown that Helmholtz problems and Poisson problems of the sort arising in atmospheric modeling can be solved efficiently and with good parallel scalability using geometric multigrid methods ( Heikes et al. 2013 ; Müller and Scheichl 2014 ; Dedner et al. 2015, manuscript submitted to Int. J. Numer. Methods Fluids ). See, for example, Fulton et al. (1986) for a clear introduction to multigrid methods in the context of atmospheric modeling. Such methods
∇ · u * ≠ 0. The Crank–Nicholson time integration in (7) (e.g., for advection), requires solving a nonlinear system [ (7) ] at each time step n + 1. A multilevel solver is proposed in this paper, where denotes a discretization of (7) at level j . The pressure equation is solved with a multigrid solver. 2) The FAS–JFNK methodology For the dynamical core of the present model, benefits of the Jacobian-free Newton–Krylov (JFNK) method (e.g., Knoll and Keyes 2004 ) are combined with that of
∇ · u * ≠ 0. The Crank–Nicholson time integration in (7) (e.g., for advection), requires solving a nonlinear system [ (7) ] at each time step n + 1. A multilevel solver is proposed in this paper, where denotes a discretization of (7) at level j . The pressure equation is solved with a multigrid solver. 2) The FAS–JFNK methodology For the dynamical core of the present model, benefits of the Jacobian-free Newton–Krylov (JFNK) method (e.g., Knoll and Keyes 2004 ) are combined with that of
optimization algorithms, and present error analyses for the optimized grids. Section 5 presents the two-dimensional multigrid solver and discusses convergence and scaling to very fine grids. A summary and conclusions are presented in section 6 . This paper is the first of a sequence. The second paper describes and presents results from a shallow-water model and a three-dimensional hydrostatic model, both based on the grid described here. The third paper describes a three-dimensional nonhydrostatic
optimization algorithms, and present error analyses for the optimized grids. Section 5 presents the two-dimensional multigrid solver and discusses convergence and scaling to very fine grids. A summary and conclusions are presented in section 6 . This paper is the first of a sequence. The second paper describes and presents results from a shallow-water model and a three-dimensional hydrostatic model, both based on the grid described here. The third paper describes a three-dimensional nonhydrostatic
1. Introduction Variable-resolution (or multiresolution) approaches to simulating regional climate offer an appealing modeling framework in which areas of interest can be simulated at high resolution while maintaining consistent dynamics and physics that are usually lacking in traditional regional climate models (e.g., Déqué et al. 1994 ; Wang et al. 2004 ; McGregor 2005 ; Fox-Rabinovitz et al. 2006 ; McGregor 2013 ). However, this very consistency highlights an ongoing modeling challenge
1. Introduction Variable-resolution (or multiresolution) approaches to simulating regional climate offer an appealing modeling framework in which areas of interest can be simulated at high resolution while maintaining consistent dynamics and physics that are usually lacking in traditional regional climate models (e.g., Déqué et al. 1994 ; Wang et al. 2004 ; McGregor 2005 ; Fox-Rabinovitz et al. 2006 ; McGregor 2013 ). However, this very consistency highlights an ongoing modeling challenge
