Derivation and Implementation of a Linear Multistep Numerical Scheme of Order 12

Abstract

A great many physical occurrences give rise to problems that often result in differential equations. When we solve a differential equation, we are in effect solving the physical problem it represents. Traditionally, solutions to differential equations were derived using analytical or exact methods. These solutions are often useful as they provide excellent insight into the behavior of some systems. However, certain differential equations are very difficult to solve by any means other than an approximate solution by the application of numerical methods. These methods can be classified into two thus: One-step and multistep methods. However, in this work, our focus is on a class of multistep methods known as the Linear Multi-step Methods. We thus use Taylor series expansion to derive a linear multistep method of order 12. The method is tested for consistency and zero-stability in order to establish its convergence. Also, provided are some examples of problems solved using the new scheme, and the results compared with exact solution.