Jules Kouatchou

Parallel implementation of a high-order implicit collocation method for the heat equation

We combine a high-order compact finite difference approximation and collocation techniques to numerically solve the two-dimensional heat equation. The resulting method is implicit and can be parallelized with a strategy that allows parallelization across both time and space. We compare the parallel implementation of the new method with a classical implicit method, namely the Crank–Nicolson method, where the parallelization is done across space only. We find the set of conditions for which each method is more advantageous than the other. Numerical experiments are carried out on the SGI Origin 2000.

Finite Differences and Collocation Methods for the Solution of the Two Dimensional Heat Equation

In this paper we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two dimensional heat equation. We employ respectively a second-order and a fourth-order schemes for the spatial derivatives and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is non-singular. Numerical experiments carried out on serial computers, show the unconditional stability of the proposed method and the high accuracy achieved by the fourth-order scheme.

Comparison of time and spatial collocation methods for the heat equation

We combine a high-order compact finite difference scheme to approximate the spatial derivatives and collocation techniques for the time component to numerically solve the two-dimensional heat equation. We use two approaches to implement the time collocation methods. The first one is based on an explicit computation of the coefficients of polynomials and the second one relies on differential quadratures. We also implement a spatial collocation method where differential quadratures are utilized for spatial derivatives and an implicit scheme for marching in time. We compare all the three techniques by studying their merits and analyzing their numerical performance. Our experiments show that all of them achieve high-accurate approximate solution but the time collocation method with differential quadrature offers (with respect to the one with explicit polynomials) less computational complexity and a better efficiency. All our computations, based on parallel algorithms, are carried out on the CRAY SV1.

A family of fourth-order difference schemes on rotated grid for two-dimensional convection–diffusion equation

We derive a family of fourth-order finite difference schemes on the rotated grid for the two-dimensional convection–diffusion equation with variable coefficients. In the case of constant convection coefficients, we present an analytic bound on the spectral radius of the line Jacobi’s iteration matrix in terms of the cell Reynolds numbers. Our analysis and numerical experiments show that the proposed schemes are stable and produce highly accurate solutions. Classical iterative methods with these schemes are convergent with large values of the convection coefficients. We also compare the fourth-order schemes with the nine point scheme obtained from the second-order central difference scheme after one step of cyclic reduction.

On cyclic reduction and finite difference schemes

We investigate a family of finite difference schemes for discretizing the two dimensional Poisson equation on both the standard and the reduced grids. We study the relation between the cyclic reduction method and the discretization schemes on different grids. The spectral radii of the Jacobi iteration matrices, and the truncation errors of different discretization schemes are compared analytically and numerically.

A dynamic injection operator in a multigrid solution of convection-diffusion equations†

SUMMARY A multigrid method is studied for the solution of a linear system resulting from the high-order nine-point discretization of the convection-diffusion equations. The residual injection operator is used as a substitute for the usual full-weighting in the multigrid process. A heuristic analysis is given to obtain a dynamic injection operator that is cost-effective for both diffusion- and convection-dominated problems. Numerical experiments are employed to test the stability and efficiency of the proposed method.

Optimal injection operator and high order schemes for multigrid solution of 3D poisson equation

We present a multigrid solution of the three dimensional Poisson equation with a fourth order 19-point compact finite difference scheme. Using a red–black ordering of the grid points and some geometric considerations, we derive an optimal scaled injection operator for the multigrid algorithm. Numerical computations show that this operator yields not only the smallest overall CPU time, but also the best convergence rate compared to other more traditional projection operators. In addition, we present a family of 19-point compact schemes and numerically show that each one has a different optimal scaled injection operator.

Chapter 5 - A Retrospective Analysis of the Pioneering Data Assimilation Experiments with the Mintz – Arakawa General Circulation Model

This chapter discusses the retrospective analysis of the pioneering data assimilation experiments with the Mintz–Arakawa general circulation model, which have a profound impact on satellite meteorology. The basic objective of observing-system simulation studies is the determination of the relationship between the temperature errors and the inferred global winds and pressures, for realistic configurations of a proposed earth observing system with advanced vertical temperature sounders. Numerical results obtained with the Goddard Earth Observing System (GEOS) general circulation models (GCM) indicate that if a continuing day-by-day sequence or history of temperature profiles is inserted into the numerical integrations at appropriate time intervals, wind components and sea-level pressure can be determined to a useful degree of accuracy. GCMs of higher spatial and vertical resolution assimilate temperature data to substantially improve the inferred winds and sea-level pressure where no data are available. Assimilating surface pressure greatly improves the rate of adjustment and the asymptotic accuracies of the extratropical winds, but does not significantly improve the inferred tropical winds.

GEOS-Chem High Performance (GCHP): A next-generation implementation of the GEOS-Chem chemical transport model for massively parallel applications

Global modeling of atmospheric chemistry is a grand computational challenge because of the need to simulate large coupled systems of ∼ 100–1000 chemical species interacting with transport on all scales. Offline chemical transport models (CTMs), where the chemical continuity equations are solved using meteorological data as input, have usability advantages and are important vehicles for developing atmospheric chemistry knowledge that can then be transferred to Earth system models. However, they have generally not been designed to take advantage of massively parallel computing architectures. Here, we develop such a highperformance capability for GEOS-Chem (GCHP), a CTM driven by meteorological data from the NASA Goddard Earth Observation System (GEOS) and used by hundreds of research groups worldwide. GCHP is a grid-independent implementation of GEOS-Chem using the Earth System Modeling Framework (ESMF) that permits the same standard model to operate in a distributed-memory framework for massive parallelization. GCHP also allows GEOS-Chem to take advantage of the native GEOS cubed-sphere grid for greater accuracy and computational efficiency in simulating transport. GCHP enables GEOS-Chem simulations to be conducted with high computational scalability up to at least 500 cores, so that global simulations of stratosphere– troposphere oxidant–aerosol chemistry at C180 spatial resolution (∼ 0.5× 0.625) or finer become routinely feasible.

The description and validation of a computationally-Efficient CH 4 -CO-OH (ECCOHv1.01) chemistry module for 3-D model applications

We present the Efficient CH4–CO–OH (ECCOH) chemistry module that allows for the simulation of the methane, carbon monoxide, and hydroxyl radical (CH4–CO–OH) system, within a chemistry climate model, carbon cycle model, or Earth system model. The computational efficiency of the module allows many multi-decadal sensitivity simulations of the CH4–CO–OH system, which primarily determines the global atmospheric oxidizing capacity. This capability is important for capturing the nonlinear feedbacks of the CH4–CO–OH system and understanding the perturbations to methane, CO, and OH, and the concomitant impacts on climate. We implemented the ECCOH chemistry module in the NASA GEOS-5 atmospheric global circulation model (AGCM), performed multiple sensitivity simulations of the CH4–CO–OH system over 2 decades, and evaluated the model output with surface and satellite data sets of methane and CO. The favorable comparison of output from the ECCOH chemistry module (as configured in the GEOS-5 AGCM) with observations demonstrates the fidelity of the module for use in scientific research.