Numerical Methods for Differential Equations

After revisiting some practical problems of numerical methods by playing with models introduced in Chapter 2, we will investigate the two most important properties of numerical algorithms—stability and accuracy. This is done most easily in the context of the simplest numerical method, i.e., the explicit Euler method (Section 5.2). The problem of solving stiff differential equations is addressed in the following section (Section 5.3), again by using Euler’s method. Here we learn about the utility of implicit methods. Higher order methods are introduced in Section 5.4, and algorithms for step-size control are briefly mentioned in Section 5.5. Finally, in Section 5.6, we shall briefly turn to numerical methods for (one-dimensional) spatial systems, i.e., finite element methods. dY x dx F x Y Y x Y o o ( ) ( , ) , ( ) = =