Search

We consider the Variational Data Assimilation (VarDA) problem in an operational framework, namely, as it results when it is employed for the analysis of temperature and salinity variations of data collected in closed and semi closed seas. We present a computing approach to solve the main computational kernel at the heart of the VarDA problem, which outperforms the technique nowadays employed by the oceanographic operative software. The new approach is obtained by means of Tikhonov regularization. We provide the sensitivity analysis of this approach and we also study its performance in terms of the accuracy gain on the computed solution. We provide validations on two realistic oceanographic data sets.

In this paper, we propose a new paradigm for solving Navier-Stokes equations. The proposed methodology is based on a streamfunction-velocity formulation of the two-dimensional steady-state Navier-Stokes equations representing incompressible fluid flows in two-dimensional domains. Similar formulations are also possible for three-dimensional fluid flows. The main advantage of our formulation is that it avoids the difficulties associated with the computation of vorticity values, especially on solid boundaries, encountered when solving the streamfunction-vorticity formulations. Our formulation also avoids the difficulties associated with solving pressure equations of the conventional velocity-pressure formulations of the Navier-Stokes equations. We describe the new formulation of the Navier-Stokes equations and use this formulation to solve a couple of fluid flow problems. We use a biconjugate gradient method to obtain the numerical solutions of the fluid flow problems and provide detailed comparison data for the lid driven cavity flow problem. It is discovered that our new formulation successfully provides high accuracy solutions for the benchmark problem. In addition, we also solve a problem of flow in a rectangular cavity with aspect ratio 2 and compare our results qualitatively and quantitatively with numerical and experimental results available in the literature. In all cases, we obtain high accuracy solutions with little additional cost.

We combine a compact high-order difference approximation with multigrid V-cycle algorithm to solve the two-dimensional Poisson equation with Dirichlet boundary conditions. This scheme, along with several different orderings of grid space and projection operators, is compared with the five-point formula to show the dramatic improvement in computed accuracy, on serial and vector machines.

We introduce a multigrid algorithm to solve the convection- diffusion equations using a nine-point compact difference scheme. We test the efficiency of the algorithm with various smoothers and intergrid transfer operators. The algorithm displays a grid-independent convergence rate and produces solutions with high accuracy. Numerical results are presented to validate the conclusions.

In recent years, high accuracy finite difference approximations were developed for partial differential equations of elliptic type, with particular emphasis on the convection-diffusion equation. These approximations are of compact type, have a local truncation error of fourth order, and allow the use of standard iterative schemes to solve the resulting systems of algebraic equations. These high accuracy approximations are extended to the solution of Navier-Stokes equations. Solutions are obtained for the model problem of driven cavity and are compared with solutions obtained using other approximations and those obtained by other authors. It is discovered that the high order approximations do indeed produce high accuracy solutions and have a potential for use in solving important problems of viscous fluid flows.

The two-dimensional Navier-Stokes equations governing the steady flow of a viscous incompressible fluid are usually written in the convective or the divergence form. There is a continuing controversy in the literature about the superiority of the divergence form over the convective form. The effect of central and upwind differencing is examined on each of these formulations and it is concluded that none of these formulations has absolute superiority over the other. The problem of the driven cavity is taken as a test model.

The way in which the boundary values of the vorticity are approximated in the numerical solution of the Navier-Stokes equations affects the rate of convergence and accuracy of the solutions. In this paper two classes of boundary approximations are studied. The problem of viscous flow in a square cavity is chosen as a model. Numerical solutions are obtained for Reynolds numbers 1, 10, 50, 100, 500 and 1000 and the iterative procedure is found to become faster with a decrease in the local accuracy of the boundary approximation. Detailed comparisons are carried out in order to determine accuracy of various numerical solutions. Several parameters, based on the numerical solutions, are found to vary monotonically and approach certain limiting values. These parameters are considered to be reliable indicators of accuracy and are recommended for comparison of numerical results obtained by different methods.

The Dirichlet problem for the biharmonic equation is solved using the thirteen-point formula. The prescribed normal derivative on the boundary is replaced by two classes of boundary approximations in order to define the solution at certain fictitious node points. A direct method is used to solve the resulting system of algebraic equations. It is found that the accuracy of the numerical solution strongly depends upon the boundary approximation used, as in the coupled-equation approach. However, the cost of obtaining the solution is almost independent of the boundary approximation, unlike the coupled-equation approach.