A Two-Stage Fourth-Order Multimoment Global Shallow-Water Model on the Cubed Sphere

2020 ◽  
Vol 148 (10) ◽  
pp. 4267-4279
Author(s):  
Yuzhang Che ◽  
Chungang Chen ◽  
Feng Xiao ◽  
Xingliang Li ◽  
Xueshun Shen
Keyword(s):  
Shallow Water ◽  
Fourth Order ◽  
Time Integration ◽  
Water Model ◽  
Shallow Water Model ◽  
Time Step ◽  
Two Stage ◽  
Order Scheme ◽  
Runge Kutta ◽  
Cubed Sphere

AbstractA new multimoment global shallow-water model on the cubed sphere is proposed by adopting a two-stage fourth-order Runge–Kutta time integration. Through calculating the values of predicted variables at half time step t = tn + (1/2)Δt by a second-order formulation, a fourth-order scheme can be derived using only two stages within one time step. This time integration method is implemented in our multimoment global shallow-water model to build and validate a new and more efficient numerical integration framework for dynamical cores. As the key task, the numerical formulation for evaluating the derivatives in time has been developed through the Cauchy–Kowalewski procedure and the spatial discretization of the multimoment finite-volume method, which ensures fourth-order accuracy in both time and space. Several major benchmark tests are used to verify the proposed numerical framework in comparison with the existing four-stage fourth-order Runge–Kutta method, which is based on the method of lines framework. The two-stage fourth-order scheme saves about 30% of the computational cost in comparison with the four-stage Runge–Kutta scheme for global advection and shallow-water models. The proposed two-stage fourth-order framework offers a new option to develop high-performance time marching strategy of practical significance in dynamical cores for atmospheric and oceanic models.

Conservative Transport Schemes for Spherical Geodesic Grids: High-Order Flux Operators for ODE-Based Time Integration

2011 ◽  
Vol 139 (9) ◽  
pp. 2962-2975 ◽  
Author(s):  
William C. Skamarock ◽  
Almut Gassmann
Keyword(s):  
Fourth Order ◽  
Time Integration ◽  
Higher Order ◽  
Second Order ◽  
Time Step ◽  
Diffusion Term ◽  
The Third ◽  
Integration Schemes ◽  
Order Scheme ◽  
Runge Kutta

Higher-order finite-volume flux operators for transport algorithms used within Runge–Kutta time integration schemes on irregular Voronoi (hexagonal) meshes are proposed and tested. These operators are generalizations of third- and fourth-order operators currently used in atmospheric models employing regular, orthogonal rectangular meshes. Two-dimensional least squares fit polynomials are used to evaluate the higher-order spatial derivatives needed to cancel the leading-order truncation error terms of the standard second-order centered formulation. Positive definite or monotonic behavior is achieved by applying an appropriate limiter during the final Runge–Kutta stage within a given time step. The third- and fourth-order formulations are evaluated using standard transport tests on the sphere. The new schemes are more accurate and significantly more efficient than the standard second-order scheme and other schemes in the literature examined by the authors. The third-order formulation is equivalent to the fourth-order formulation plus an additional diffusion term that is proportional to the Courant number. An optimal value for the coefficient scaling this diffusion term is chosen based on qualitative evaluation of the test results. Improvements using the higher-order scheme in place of the traditional second-order centered approach are illustrated within 3D unstable baroclinic wave simulations produced using two global nonhydrostatic models employing spherical Voronoi meshes.

scholarly journals A Discontinuous Galerkin Global Shallow Water Model

2005 ◽  
Vol 133 (4) ◽  
pp. 876-888 ◽  
Author(s):  
Ramachandran D. Nair ◽  
Stephen J. Thomas ◽  
Richard D. Loft
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Time Integration ◽  
Curvilinear Coordinates ◽  
Basis Set ◽  
Water Model ◽  
Test Case ◽  
Shallow Water Model ◽  
Cubed Sphere

A discontinuous Galerkin shallow water model on the cubed sphere is developed, thereby extending the transport scheme developed by Nair et al. The continuous flux form nonlinear shallow water equations in curvilinear coordinates are employed. The spatial discretization employs a modal basis set consisting of Legendre polynomials. Fluxes along the element boundaries (internal interfaces) are approximated by a Lax–Friedrichs scheme. A third-order total variation diminishing Runge–Kutta scheme is applied for time integration, without any filter or limiter. Numerical results are reported for the standard shallow water test suite. The numerical solutions are very accurate, there are no spurious oscillations in test case 5, and the model conserves mass to machine precision. Although the scheme does not formally conserve global invariants such as total energy and potential enstrophy, conservation of these quantities is better preserved than in existing finite-volume models.

Diffusion Experiments with a Global Discontinuous Galerkin Shallow-Water Model

2009 ◽  
Vol 137 (10) ◽  
pp. 3339-3350 ◽  
Author(s):  
Ramachandran D. Nair
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Water Model ◽  
Order System ◽  
Small Scale ◽  
Shallow Water Model ◽  
First Order ◽  
Advection Diffusion Equation ◽  
Diffusion Scheme ◽  
Cubed Sphere

Abstract A second-order diffusion scheme is developed for the discontinuous Galerkin (DG) global shallow-water model. The shallow-water equations are discretized on the cubed sphere tiled with quadrilateral elements relying on a nonorthogonal curvilinear coordinate system. In the viscous shallow-water model the diffusion terms (viscous fluxes) are approximated with two different approaches: 1) the element-wise localized discretization without considering the interelement contributions and 2) the discretization based on the local discontinuous Galerkin (LDG) method. In the LDG formulation the advection–diffusion equation is solved as a first-order system. All of the curvature terms resulting from the cubed-sphere geometry are incorporated into the first-order system. The effectiveness of each diffusion scheme is studied using the standard shallow-water test cases. The approach of element-wise localized discretization of the diffusion term is easy to implement but found to be less effective, and with relatively high diffusion coefficients, it can adversely affect the solution. The shallow-water tests show that the LDG scheme converges monotonically and that the rate of convergence is dependent on the coefficient of diffusion. Also the LDG scheme successfully eliminates small-scale noise, and the simulated results are smooth and comparable to the reference solution.

scholarly journals Forcing for a Cascaded Lattice Boltzmann Shallow Water Model

Water ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 439 ◽  
Author(s):  
Sara Venturi ◽  
Silvia Di Francesco ◽  
Martin Geier ◽  
Piergiorgio Manciola
Keyword(s):  
Shallow Water ◽  
Lattice Boltzmann ◽  
Model Performance ◽  
Lattice Boltzmann Model ◽  
Water Model ◽  
Two Dimensional ◽  
Shallow Water Model ◽  
Basic Scheme ◽  
Order Scheme ◽  
Boltzmann Model

This work compares three forcing schemes for a recently introduced cascaded lattice Boltzmann shallow water model: a basic scheme, a second-order scheme, and a centred scheme. Although the force is applied in the streaming step of the lattice Boltzmann model, the acceleration is also considered in the transformation to central moments. The model performance is tested for one and two dimensional benchmarks.

scholarly journals Congested shallow water model: roof modeling in free surface flow

2018 ◽  
Vol 52 (5) ◽  
pp. 1679-1707 ◽  
Author(s):  
Edwige Godlewski ◽  
Martin Parisot ◽  
Jacques Sainte-Marie ◽  
Fabien Wahl
Keyword(s):  
Shallow Water ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Hyperbolic Equations ◽  
Mechanical Energy ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Water Model ◽  
Shallow Water Model ◽  
Time Step

We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a finite volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.

Comparing Numerical Accuracy of Icosahedral A-Grid and C-Grid Schemes in Solving the Shallow-Water Model

2020 ◽  
Vol 148 (10) ◽  
pp. 4009-4033
Author(s):  
Yonggang G. Yu ◽  
Ning Wang ◽  
Jacques Middlecoff ◽  
Pedro S. Peixoto ◽  
Mark W. Govett
Keyword(s):  
Energy Conservation ◽  
Shallow Water ◽  
Time Integration ◽  
Grid Method ◽  
Maximum Error ◽  
Water Model ◽  
Shallow Water Model ◽  
Numerical Accuracy ◽  
Specific Test ◽  
Computational Performance

AbstractA single software framework is introduced to evaluate numerical accuracy of the A-grid (NICAM) versus C-grid (MPAS) shallow-water model solvers on icosahedral grids. The C-grid staggering scheme excels in numerical noise control and total energy conservation, which results in exceptional stability for long time integration. Its weakness lies in the lack of model error reduction with increasing resolution in specific test cases (especially the root-mean-square error). The A-grid method conserves well potential enstrophy and shows a linear reduction of error with increasing resolution. The gridpoint noise manifests itself clearly on A-grid, but much less on C-grid. We show that the Coriolis force term on C-grid has a larger error than on A-grid. To treat the Coriolis term and kinetic energy gradient on an equal footing on C-grid, we propose combining these two quantities into a single tendency term and computing its value by a linear combination operation. This modification alone reduces numerical errors but still fails to converge the maximum error with resolution. The method of Peixoto can solve the maximum-error nonconvergence problem on C-grid but degrades the numerical stability. For the steady-state thin-layer test (0.01 m in depth), the A-grid method is less susceptible than C-grid methods, which are presumably disrupted by the Hollingsworth instability. The effect of horizontal diffusion on model accuracy and energy conservation is shown in detail. Programming experience shows that software implementation and optimization can strongly influence computational performance for models, although memory requirement and computational load of the two schemes are comparable.

scholarly journals A global finite-element shallow-water model supporting continuous and discontinuous elements

2014 ◽  
Vol 7 (6) ◽  
pp. 3017-3035 ◽  
Author(s):  
P. A. Ullrich
Keyword(s):  
Finite Element ◽  
Shallow Water ◽  
Wave Train ◽  
Correction Function ◽  
Water Model ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Discontinuous Elements ◽  
Cubed Sphere

Abstract. This paper presents a novel nodal finite-element method for either continuous and discontinuous elements, as applied to the 2-D shallow-water equations on the cubed sphere. The cornerstone of this method is the construction of a robust derivative operator that can be applied to compute discrete derivatives even over a discontinuous function space. A key advantage of the robust derivative is that it can be applied to partial differential equations in either a conservative or a non-conservative form. However, it is also shown that discontinuous penalization is required to recover the correct order of accuracy for discontinuous elements. Two versions with discontinuous elements are examined, using either the g1 and g2 flux correction function for distribution of boundary fluxes and penalty across nodal points. Scalar and vector hyperviscosity (HV) operators valid for both continuous and discontinuous elements are also derived for stabilization and removal of grid-scale noise. This method is validated using four standard shallow-water test cases, including geostrophically balanced flow, a mountain-induced Rossby wave train, the Rossby–Haurwitz wave and a barotropic instability. The results show that although the discontinuous basis requires a smaller time step size than that required for continuous elements, the method exhibits better stability and accuracy properties in the absence of hyperviscosity.

scholarly journals Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

2021 ◽  
Author(s):  
Jan Ackmann ◽  
Peter Düben ◽  
Tim Palmer ◽  
Piotr Smolarkiewicz
Keyword(s):  
Machine Learning ◽  
Shallow Water ◽  
Water Model ◽  
Implicit Time ◽  
Learning Approaches ◽  
Shallow Water Model ◽  
Time Step ◽  
Linear Solvers ◽  
Linear Solver ◽  
Time Stepping

<p>Semi-implicit grid-point models for the atmosphere and the ocean require linear solvers that are working efficiently on modern supercomputers. The huge advantage of the semi-implicit time-stepping approach is that it enables large model time-steps. This however comes at the cost of having to solve a computationally demanding linear problem each model time-step to obtain an update to the model’s pressure/fluid-thickness field. In this study, we investigate whether machine learning approaches can be used to increase the efficiency of the linear solver.</p><p>Our machine learning approach aims at replacing a key component of the linear solver—the preconditioner. In the preconditioner an approximate matrix inversion is performed whose quality largely defines the linear solver’s performance. Embedding the machine-learning method within the framework of a linear solver circumvents potential robustness issues that machine learning approaches are often criticized for, as the linear solver ensures that a sufficient, pre-set level of accuracy is reached. The approach does not require prior availability of a conventional preconditioner and is highly flexible regarding complexity and machine learning design choices.</p><p>Several machine learning methods of different complexity from simple linear regression to deep feed-forward neural networks are used to learn the optimal preconditioner for a shallow-water model with semi-implicit time-stepping. The shallow-water model is specifically designed to be conceptually similar to more complex atmosphere models. The machine-learning preconditioner is competitive with a conventional preconditioner and provides good results even if it is used outside of the dynamical range of the training dataset.</p>

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