Discretization

About: Discretization is a research topic. Over the lifetime, 53069 publications have been published within this topic receiving 1077475 citations.

Navier-Stokes Equations: Theory and Numerical Analysis

Abstract: I. The Steady-State Stokes Equations . 1. Some Function Spaces. 2. Existence and Uniqueness for the Stokes Equations. 3. Discretization of the Stokes Equations (I). 4. Discretization of the Stokes Equations (II). 5. Numerical Algorithms. 6. The Penalty Method. II. The Steady-State Navier-Stokes Equations . 1. Existence and Uniqueness Theorems. 2. Discrete Inequalities and Compactness Theorems. 3. Approximation of the Stationary Navier-Stokes Equations. 4. Bifurcation Theory and Non-Uniqueness Results. III. The Evolution Navier-Stokes Equations . 1. The Linear Case. 2. Compactness Theorems. 3. Existence and Uniqueness Theorems. (n < 4). 4. Alternate Proof of Existence by Semi-Discretization. 5. Discretization of the Navier-Stokes Equations: General Stability and Convergence Theorems. 6. Discretization of the Navier-Stokes Equations: Application of the General Results. 7. Approximation of the Navier-Stokes Equations by the Projection Method. 8. Approximation of the Navier-Stokes Equations by the Artificial Compressibility Method. Appendix I: Properties of the Curl Operator and Application to the Steady-State Navier-Stokes Equations. Appendix II. (by F. Thomasset): Implementation of Non-Conforming Linear Finite Elements. Comments.

The immersed boundary method

Abstract: This paper is concerned with the mathematical structure of the immersed boundary (IB) method, which is intended for the computer simulation of fluid–structure interaction, especially in biological fluid dynamics. The IB formulation of such problems, derived here from the principle of least action, involves both Eulerian and Lagrangian variables, linked by the Dirac delta function. Spatial discretization of the IB equations is based on a fixed Cartesian mesh for the Eulerian variables, and a moving curvilinear mesh for the Lagrangian variables. The two types of variables are linked by interaction equations that involve a smoothed approximation to the Dirac delta function. Eulerian/Lagrangian identities govern the transfer of data from one mesh to the other. Temporal discretization is by a second-order Runge–Kutta method. Current and future research directions are pointed out, and applications of the IB method are briefly discussed. Introduction The immersed boundary (IB) method was introduced to study flow patterns around heart valves and has evolved into a generally useful method for problems of fluid–structure interaction. The IB method is both a mathematical formulation and a numerical scheme. The mathematical formulation employs a mixture of Eulerian and Lagrangian variables. These are related by interaction equations in which the Dirac delta function plays a prominent role. In the numerical scheme motivated by the IB formulation, the Eulerian variables are defined on a fixed Cartesian mesh, and the Lagrangian variables are defined on a curvilinear mesh that moves freely through the fixed Cartesian mesh without being constrained to adapt to it in any way at all.

Multi-Interval Discretization of Continuous-Valued Attributes for Classification Learning

Abstract: Since most real-world applications of classification learning involve continuous-valued attributes, properly addressing the discretization process is an important problem. This paper addresses the use of the entropy minimization heuristic for discretizing the range of a continuous-valued attribute into multiple intervals.

Efficient Inverse Modeling of Barotropic Ocean Tides

Abstract: A computationally efficient relocatable system for generalized inverse (GI) modeling of barotropic ocean tides is described. The GI penalty functional is minimized using a representer method, which requires repeated solution of the forward and adjoint linearized shallow water equations (SWEs). To make representer computations efficient, the SWEs are solved in the frequency domain by factoring the coefficient matrix for a finite-difference discretization of the second-order wave equation in elevation. Once this matrix is factored representers can be calculated rapidly. By retaining the first-order SWE system (defined in terms of both elevations and currents) in the definition of the discretized GI penalty functional, complete generality in the choice of dynamical error covariances is retained. This allows rational assumptions about errors in the SWE, with soft momentum balance constraints (e.g., to account for inaccurate parameterization of dissipation), but holds mass conservation constraints. Wh...