Convergence of Alternant Theta-Method with Applications
In this work the alternant theta-method and its application is investigated. We analyze the local approximation error and the convergence of the method on the non-equidistant mesh. We define the order of convergence, as well. The main idea of this approach is the approximation of the solution of the Cauchy problems by using different numerical schemes (implicit, explicit, IMEX, one-step, multi-step etc.) with varying step-sizes. Benefits of such approximations are shown for the problems with non-smooth solutions. We show that the convergence and the error estimation can be given relatively easily for the classical \(\theta \)-method in case both equidistant and non-equidistant time/space discretizations. We analyze the connection of this approach to the classical discrete Gronwall lemma. We show that the extended discrete Gronwall lemma can be successfully applied to the estimation of the convergence’s rate of the alternant \(\theta _i\) method. We show numerical examples for some non-linear time dependent differential equations, which have non-continuous or strongly oscillated solutions.
KeywordsAlternant theta method Stability constant Gronwall lemma
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