Design of Runge-Kutta based split-explicit time integration algorithms for the NEMO ocean model

<p>The NEMO ocean model is currently based on the Leapfrog scheme that provides a good combination between simplicity and efficiency for low-resolution global simulations. However, this scheme is subject to difficulties that question its relevance at high-resolution : the necessary damping of its computational mode, e.g. via a Robert-Asselin filter, affect stability and increases amplitude and phase errors of the physical mode ; because it is unconditionally unstable for diffusive processes, monotonicity or positive-definiteness comes at a substantial cost and complication. The evolution toward a 2-level time stepping algorithm based on Runge-Kutta schemes is studied. Special attention is given to how to articulate a mode-splitting technique to handle the fast dynamics associated with the free surface. Linear stability analyses of several Runge-Kutta based, split-explicit algorithms are performed and the most promising ones are identified. They allow a good compromise between robustness, stability and accuracy for integration of internal gravity waves, Coriolis and advection processes. Idealized test-cases illustrate the benefits associated to the revised time-stepping compared to the original Leapfrog.</p>

ON A STUDY OF STAGGERED LEAPFROG SCHEME FOR LINEAR SHALLOW WATER EQUATIONS

In this paper, we present an analytical and numerical study of staggered leapfrog scheme for linear shallow water equation. It is shown that the scheme is stable when Courant number < 1, has second order accurate in both time and space, and there is no damping error in this scheme. We implement the scheme to simulate standing wave in a closed basin to show that the surface motions stay zero in a node and have constant amplitude at the antinode. For an external force given into the basin, it will induce a resonance, which cause the wave amplitude is getting bigger at the position of antinode. Moreover, we simulate a wave in a tidal basin, and show that the model has infinite spin up time. For a linear shallow water equation with linear friction, it is shown that the model has finite spin up time.

EM modelling of arbitrary shaped dispersive chiral dielectric objects using a 3D leapfrog scheme on unstructured meshes

The standard Yee FDTD algorithm is widely used in computational electromagnetics because of its simplicity and divergence free nature. A generalization of this classical scheme to 3D unstructured co-volume meshes is adopted, based on the use of a Delaunay primal mesh and its high quality Voronoi dual. This circumvents the problem of accuracy losses, which are normally associated with the use of a staircased representation of curved material interfaces in the standard Yee scheme. The procedure has been successfully employed for modelling problems involving both isotropic and anisotropic lossy materials. Here, we consider the novel extension of this approach to allow for the challenging modelling of chiral materials, where the material parameters are frequency dependent. To adequately model the dispersive behaviour, the Z-transform is employed, using second order Padé approximations to maintain the accuracy of the basic scheme. To validate the implementation, the numerical results produced are compared with available analytical solutions. The stability of the chiral algorithm is also studied.

Upwind and standard leapfrog difference schemes

Для решения задачи переноса в статье предложено использовать схему, построенную на основе линейной комбинации разностной схемы кабаре (англ. Upwind Leapfrog) и крест (англ. Standard Leapfrog) с весовыми коэффициентами, полученными в результате минимизации погрешности аппроксимации. Проведено сравнение расчетов для задачи переноса на основе предложенной схемы с результатами, полученными с использованием схемы, построенной на основе линейной комбинации схемы с центральными разностями и схемы кабаре, и двухпараметрической разностной схемы третьего порядка точности. In order to solve the transfer problem, it is proposed to use the scheme based on a linear combination of the upwind and standard leapfrog difference schemes with weighting coefficients obtained by minimizing the approximation error. The estimate of the approximation error of the proposed difference scheme shows that, for small Courant numbers, this scheme whose approximation error is $O(ch^2)$, where the constant $c$ is significantly less than unity, is preferable to use than the original upwind and standard leapfrog schemes whose approximation errors are $O(h^2)$. The numerical results for the transfer problem based on the proposed scheme are compared with the results obtained using the following schemes: (i) the scheme based on a linear combination of the standard leapfrog scheme and the upwind leapfrog sscheme and (ii) the two-parameter difference scheme of the third order of accuracy.

Enhanced biDimensional pIc: an electrostatic/magnetostatic particle-in-cell code for plasma based systems

EDI (enhanced biDimensional pIc) is a two-dimensional (2-D) electrostatic/magnetostatic particle-in-cell (PIC) code designed to optimize plasma based systems. The code is built on an unstructured mesh of triangles, allowing for arbitrary geometries. The PIC core is comprised of a Boris leapfrog scheme that can manage multiple species. Particle tracking locates particles in the mesh, using a fast and simple priority-sorting algorithm. A magnetic field with an arbitrary topology can be imposed to study the magnetized particle dynamics. The electrostatic fields are then computed by solving Poisson’s equation with a a finite element method solver. The latter is an external solver that has been properly modified in order to be integrated into EDI. The major advantage of using an external solver directly incorporated into the EDI structure is its strong flexibility, in fact it is possible to couple together different physical problems (electrostatic, magnetostatic, etc.). EDI is written in C, which allows the rapid development of new modules. A big effort in the development of the code has been made in optimization of the linking efficiency, in order to minimize computational time. Finally, EDI is a multiplatform (Linux, Mac OS X) software.