Download PDF COMPUTATIONAL MATHEMATICS Models, Methods, and Analysis with MATLAB and MPI by ROBERT E. WHITE

Sinopsis

Computational science is a blend of applications, computations and mathematics. It is a mode of scientific investigation that supplements the traditional laboratory and theoretical methods of acquiring knowledge. This is done by formulating mathematical models whose solutions are approximated by computer simulations. By making a sequence of adjustments to the model and subsequent computations one can gain some insights into the application area under consideration. This text attempts to illustrate this process as a method for scientific investigation. Each section of the first six chapters is motivated by a particular application, discrete or continuous model, numerical method, computer implementation and an assessment of what has been done.
 

Applications include heat di􀀞usion to cooling fins and solar energy storage, pollutant transfer in streams and lakes, models of vector and multiprocessing computers, ideal and porous fluid flows, deformed membranes, epidemic models with dispersion, image restoration and value of American put option contracts. The models are initially introduced as discrete in time and space, and this allows for an early introduction to partial di􀀞erential equations. The discrete models have the form of matrix products or linear and nonlinear systems. Methods include sparse matrix iteration with stability constraints, sparse matrix solutions via variation on Gauss elimination, successive over-relaxation, conjugate gradient, and minimum residual methods. Picard and Newton methods are used to approximate the solution to nonlinear systems. Most sections in the first five chapters have MATLAB R° the very a􀀞ordable current student version of MATLAB. They are intended to be studied and not used as a "black box." The MATLAB codes should be used as a first step towards more sophisticated numerical modeling. These codes do provide a learning by doing environment. The exercises at the end of each section have three categories: routine computations, variation of models, and mathematical analysis. The last four chapters focus on multiprocessing algorithms, which are implemented using message passing interface, MPI; see Rocks" cluster software. These chapters have elementary Fortran 9x codes to illustrate the basic MPI subroutines, and the applications of the previous chapters are revisited from a parallel implementation perspective.

Content

1. Discrete Time-Space Models
2. Steady State Discrete Models
3. Poisson Equation Models
4. Nonlinear and 3D Models
5. Epidemics, Images and Money
6. High Performance Computing
7. Message Passing Interface
8. Classical Methods for Ax = d
9. Krylov Methods for Ax = d

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