Recursive solution of initial value problems with temporal discretization

Abbas Edalat, Amin Farjudian, Yiran Li

Research output: Journal PublicationArticlepeer-review


We construct a continuous domain, as a model of interval analysis, for temporal discretization of differential equations. By using this domain, and the domain of Lipschitz maps, we formulate a generalization of the Euler operator, which exhibits second-order convergence. We prove computability of the operator within the framework of effectively given domains. The operator only requires the vector field of the differential equation to be Lipschitz continuous, in contrast to the related operators in the literature which require the vector field to be at least continuously differentiable. Within the same framework, we also analyze temporal discretization and computability of another variant of the Euler operator formulated according to Runge-Kutta theory. We prove that, compared with this variant, the second-order operator that we formulate directly, not only imposes weaker assumptions on the vector field, but also exhibits superior convergence rate. We implement the first-order, second-order, and Runge-Kutta Euler operators using arbitrary-precision interval arithmetic, and report on some experiments. The experiments confirm our theoretical results. In particular, we observe the superior convergence rate of our second-order operator compared with the Runge-Kutta Euler and the common (first-order) Euler operators.

Original languageEnglish
Article number114221
JournalTheoretical Computer Science
Publication statusPublished - 20 Nov 2023
Externally publishedYes


  • Domain theory
  • Euler method
  • Interval analysis
  • Lipschitz vector field
  • Runge-Kutta method

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science (all)


Dive into the research topics of 'Recursive solution of initial value problems with temporal discretization'. Together they form a unique fingerprint.

Cite this