Abstract
A new family of multistage embedded explicit multistep algorithms with the highest possible algebraic order is proposed in this paper. The newly introduced methods can be applied to the approximate solution of second order initial and/or boundary-value problems with periodical and/or oscillating solutions.
Requiring phase-lag = 0 and phase-lag((n)) = 0, where phase-lag((n)) is the n-th derivative of the phase-lag, for the algorithms of the above family, a system of of equations is produced. The solution of the above obtained system of equations leads to the determination of the free parameters of the new introduced schemes.
A theoretical analysis (local truncation error and stability analysis) follows the construction of the new family of algorithms. The theoretical analysis consists of the following stages:
Determination of the local truncation error (LTE) of the specific algorithm of the newly introduced family.
Study of the asymptotic form of the LTE which is produced using the radial time independent Schrodinger equation.
Evaluation of the asymptotic forms of LTEs for the methods of the newly introduced family. The above-mentioned evaluation leads to conclusions on the efficiency of the schemes of the newly introduced family.
Study of the stability and the interval of the periodicity of the schemes of the newly introduced family.
Study of the efficiency of the schemes of the newly introduced family, by applying them to the approximate solution of numerous second order problems like the radial Schrodinger equation, astronomical problems, etc. The above presented numerical experiments lead to conclusions on the effectiveness of the schemes.