Sinopsis
The central topic of this book is the programming and use of a set of library routines for the numerical solution (integration) of systems of initial value ordinary differential equations (ODEs). We start by reviewing some of the basic concepts of ODEs, including methods of integration, that are the mathematical foundation for an understanding of the ODE integration routines.
Ideally, we would like to have a higher-order ODE integration method (higher than the first-order Euler method) without having to take derivatives of the ODEs. Although this may seem like an impossibility, it can in fact be done by the Runge Kutta (RK) method. In other words, the RK method can be used to fit the numerical ODE solution exactly to an arbitrary number of terms in the underlying Taylor series without having to differentiate the ODE. We will investigate theRKmethod, which is the basis for theODEintegration routines described in this book.
The other important characteristic of a numerical integration algorithm (in addition to not having to differentiate the ODE) is a way of estimating the truncation error, ε, so that the integration step, h, can be adjusted to achieve a solution with a prescribed accuracy. This may also seem like an impossibility since it would appear that in order to compute ε we need to know the exact (analytical) solution. But if the exact solution is known, therewouldbe no need to calculate the numerical solution. The answer to this apparent contradiction is the fact that we will calculate an estimate of the truncation error (and not the exact truncation error which would imply thatweknowthe exact solution).
Content
- Some Basics of ODE Integration
- Solution of a 1x1 ODE System
- Solution of a 2x2 ODE System
- Solution of a Linear PDE
- Solution of a Nonlinear PDE
