  In this paper, we construct a novel Eulerian-Lagrangian finite volume (ELFV)
method for nonlinear scalar hyperbolic equations in one space dimension. It is
well known that the exact solutions to such problems may contain shocks though
the initial conditions are smooth, and direct numerical methods may suffer from
restricted time step sizes. To relieve the restriction, we propose an ELFV
method, where the space-time domain was separated by the partition lines
originated from the cell interfaces whose slopes are obtained following the
Rakine-Hugoniot junmp condition. Unfortunately, to avoid the intersection of
the partition lines, the time step sizes are still limited. To fix this gap, we
detect effective troubled cells (ETCs) and carefully design the influence
region of each ETC, within which the partitioned space-time regions are merged
together to form a new one. Then with the new partition of the space-time
domain, we theoretically prove that the proposed first-order scheme with Euler
forward time discretization is total-variation-diminishing and
maximum-principle-preserving with {at least twice} larger time step constraints
than the classical first order Eulerian method for Burgers' equation. Numerical
experiments verify the optimality of the designed time step sizes.
