Computational Science with Mathematica

12 Easy Pieces: Essential Operations in Computational Science

James L. Noyes

Wittenberg University

Abstract

Computational Science (COSC) is a scientific investigation of physical processes on the computer by the use of computational models and methods.  Hypotheses are often formulated using mathematical models that can then be used to compute quantities of scientific interest and to gain new insights.  COSC is now on par with the development of scientific theory and the use of experimentation in order to understand the real world.  Mathematica is an ideal tool to use in this endeavor and also to teach the essential types of operations.  These types of operations may be placed into three categories: Evaluation, Simulation, and Optimization.  Within each category, fundamental operations will be described. One or more examples will be presented for each operation.  This material is currently used to teach an undergraduate course in COSC.

Evaluation: This is the most basic category since the operations in this category are used in both simulation and optimization.

1.   Analytic (symbolic) versus numeric computation
2.   Data input/output and visualization
3.   Physical units and conversions
4.   Linear versus nonlinear systems
5.   Data and function approximation
6.   Computer arithmetic
7.   Uncertainty and sensitivity in computation

Simulation: This is the most general category where models must often be implemented by writing application-specific programs.

8.   Discrete equation models
9.   Differential and integral equation models
10. Monte Carlo models

Optimization: This important category contains models that are designed to produce the best solution and typically require many controlled evaluations to determine a minimum or maximum.

11. Continuous optimization models
12. Discrete optimization models