1.3.1 Errors
There are two sources of error when we solve an ODE with a numerical method. The first is rounding error, due to the finite precision of floating-point arithmetic. The second is truncation error (sometimes called discretization error), due to the method used. The truncation error would remain even if we were to use exact arithmetic. We will be concerned with truncation error, which is typically the dominant error source for our problems of interest. We will further define two types of truncation error: the global error is the difference between the computed solution at \(t^ n\) and the true solution of the ODE at \(t^ n\), while the local error is the error made in one step of the numerical method, i.e. the difference between the computed solution at \(t^ n\) and the solution of the ODE passing through the previous point \((t^{n-1},v^{n-1})\). We defer the global error to Section 1.4.2 and will investigate the local error here.