These notes are a variation of a course given at North Carolina State University (NCSU) by R. E. White. At NCSU these notes are supplemented by "Getting Started" material from the North Carolina Supercomputing Center, Matlab and Maple. In the following notes we have tried to remain independent of the choice of computing tools, and in most sections there are several implementations of the algorithms.
The notes are intended to bridge the gap between the first year "programming" course and a full year course on scientific computing and numerical analysis such as given in "Numerical Analysis" by Burden and Faires. The topics are all problem driven, and number of application areas are used with an emphasis on heat and mass transfer problems. Each section or "lecture" starts with some introductory material and an application area. The mathematical model and a method of computation are described. Several computer implementations are presented. An assessment and comparison of each of the above stages is given. There are a variety of homework problems.
There are three chapters, and each chapter could be viewed as a one credit course where each section requires one or two 50 minute lectures. The first chapter is called Matrix Computations, and it is concerned with matrix models of the form newy = A oldy + d and Ax = d where A is a matrix, and d, newy and oldy are columns vectors. Calculus is not required, and only some matrices and systems of equations are required. The second chapter is on Computational Calculus, and it is concerned with the first and second order convergence of numerical integration, ordinary differential equations and fixed point problems. The student should have had the first semester of calculus and be at least taking the second semester of calculus. The first chapter is not required. The third chapter is on Numerical Solution of Boundary Value Problems, and it requires the previous two chapters and the student should be at least taking the third semester of calculus. The objective is to introduce and to find numerical solutions of two and three dimensional heat and mass transfer models.
Introduction:
Computational Science
1.2 Computers and Matrix Models
1.3 Vector Computers and Heat Diffusion
1.4 Multiprocessor Computing and Heat Transfer
1.5 Triangular Algebraic Systems
1.6 Gaussian Elimination and Steady State Heat Conduction
1.7 Ill-conditioned Algebraic Systems and Function Approximation
1.8 Overdetermined Systems and Curve Fitting to Data
2.2 Numerical Integration and the Mean Value Theorem
2.3 Numerical Integration and the Trapezoid Rule
2.4 Initial Value Problems: Euler's Method
2.5 Stiff Initial Value Problems
2.6 Fixed Points and Picard's Algorithm
2.7 Roots and Newton's Algorithm
2.8 Taylor Polynomials and Approximations
3.2 Efficient Numerical Solution of Tridiagonal Algebraic Systems
3.3 Iterative Methods: Jacobi and SOR
3.4 Block Tridiagonal Matrices and Diffusion in 2D
3.5 Fluid Flow in 2D
3.6 Time Dependent 2D Problems
3.7 Nonlinear Boundary Value Problems and Radiative Heat Transfer
3.8 High Performance Computing and Diffusion in 3D
Copyright ©1994, R. E. White. All rights reserved.