Global error evaluation strategies in multistep methods applied to ordinary differential equations and index 1 differential algebraic systems
Differential Equations (DEs) are among the most widely used mathematical tools in different area of sciences. Solving DEs, either analytically or numerically, has become a centre of interest for many mathematicians and a large variety of methods are nowadays available to solve DEs numerically. When solving a mathematical problem numerically, evaluating the error is of high importance in practice. Most of the methods already available for solving DEs are implemented with a mechanism to perform a local error control. However, in the real realm, it is common to require the numerical solution to approximate the exact solution with accuracy to a certain number of decimal places or significant figures. To satisfy this condition, we require the global error to be bounded by a specifically determined tolerance. In this case, a local error control is not longer efficient. On one hand, controlling the local error only cannot ensure that the required accuracy will be achieved. On the other hand, the use of such approach requires the user to do some preliminary studies on the problem, and have deep understanding of the method. Thus, we need a mechanism to control the global error in order to compute the numerical solution for a user-supplied accuracy requirement in automatic mode. The global error estimate calculated in the course of such a control can also be applied to improve the numerical solution obtained. It is straight forward since, if the error estimate is found with sufficiently high accuracy, we can just add it to the numerical solution to get a better approximation to the exact value. Thus, accurate evaluation of the the global error is crucial for the purpose mentioned above. Several techniques are already developed to compute the global error of the numerical solution. The most common algorithms include the Richardson extrapolation, Zadunaisky’s technique, Solving for the correction, and Using two different methods. These methods use two integrations to evaluate the global error, and the provided error estimate is valid if the global error admits an expansion in powers of the step size. Another approach, known as solving the linearised discrete variational equation, can also be used. This last differs from the others by the use of a truncated Taylor expansion of the defect of the method to estimate the global error; and solving the problem and estimating the error is roughly the same as one step of the underlying method. In this research, we will investigate numerically and compare the efficiency of different techniques for global error evaluation applied to multistep methods for solving ordinary differential equations (ODEs) and differential algebraic equations (DAEs). We will first study the global error evaluation techniques in multistep formulas for solving ODEs on uniform grids. In the case of nonuniform grids, both multistep methods with variable coefficients and interpolation-type multistep methods will be considered. Then, we will extend our study to multistep methods for solving DAEs. Theoretical background will accompany numerical works. The accuracy and reliability of the global error evaluation strategies will be discussed and compared for different types of multistep methods for solving ODEs and DAEs. We will analyse the efficiency in terms of accuracy obtained and CPU time spent. For that, a series of numerical experiments is conducted on a set of test problems with known solutions.