  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.
