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A vast amount of computer models are based on ordinary and partial differential equations. This book is an introduction to the various scientific ingredients we need for reliable computing with such type of models. A key theme is to solve differential equations on a computer. Many methods are available for this purpose, but the focus here is on , because these are simple, yet versatile, for solving a wide range of ordinary and partial differential equations. The present chapter first presents the mathematical ideas of finite difference methods and derives algorithms, i.e., formulations of the methods ready for computer programming. Then we create programs and learn how we can be sure that the programs really work correctly.

We investigate in detail the numerical quality of solving this problem by various methods, including the Forward Euler, Backward Euler, and Crank-Nicolson schemes. The analysis is based on comparing the numerical and exact solutions analytically.

The example programs associated with this chapter are found in the directory src/analysis.

This chapters targets generalizations of the simple decay model and also to look at additional numerical solution methods. We consider first variable coefficients, , and later a completely general scalar ODE and its generalization to a system of such general ODEs. Among numerical methods, we treat implicit multi-step methods, and several families of explicit methods: Leapfrog schemes, Runge–Kutta methods, and Adams–Bashforth formulas.

This chapter presents many mathematical models that all end up with ODEs of the type . The applications are taken from biology, finance, and physics, and cover population growth or decay, interacting predator-prey populations, compound interest and inflation, radioactive decay, chemical and biochemical reaction, spreading of diseases, cooling of objects, compaction of geological media, pressure variations in the atmosphere, viscoelastic response in materials, and air resistance on falling or rising bodies.

Teaching material on scientific computing has traditionally been very focused on the mathematics and the applications, while details on how the computer is programmed to solve the problems have received little attention. Many end up writing as simple programs as possible, without being aware of much useful computer science technology that would increase the fun, efficiency, and reliability of the their scientific computing activities.

This chapter demonstrates a series of good practices and tools from modern computer science, using the simple mathematical problem , , such that we minimize the mathematical details and can go more in depth with implementations. The goal is to increase the technological quality of computer programming and make it match the more well-established quality of the mathematics of scientific computing.

The conventions and techniques outlined here will save you a lot of time when you incrementally extend software over time from simpler to more complicated problems. In particular, you will benefit from many good habits: