Recursive Solution of Initial Value Problems with Temporal Discretization

TL;DR

This paper introduces a new second-order Euler operator for solving initial value problems with weaker assumptions on the vector field, demonstrating superior convergence through theoretical analysis and experiments.

Contribution

It formulates a generalized second-order Euler operator with weaker conditions and proves its computability and superior convergence compared to existing methods.

Findings

The second-order operator exhibits second-order convergence.

The operator requires only Lipschitz continuity of the vector field.

Experiments confirm the theoretical convergence advantages.

Abstract

We construct a continuous domain 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.