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In this paper, we construct a novel Eulerian-Lagrangian finite volume (ELFV)method for nonlinear scalar hyperbolic equations in one space dimension. It iswell known that the exact solutions to such problems may contain shocks thoughthe initial conditions are smooth, and direct numerical methods may suffer fromrestricted time step sizes. To relieve the restriction, we propose an ELFVmethod, where the space-time domain was separated by the partition linesoriginated from the cell interfaces whose slopes are obtained following theRakine-Hugoniot junmp condition. Unfortunately, to avoid the intersection ofthe partition lines, the time step sizes are still limited. To fix this gap, wedetect effective troubled cells (ETCs) and carefully design the influenceregion of each ETC, within which the partitioned space-time regions are mergedtogether to form a new one. Then with the new partition of the space-timedomain, we theoretically prove that the proposed first-order scheme with Eulerforward time discretization is total-variation-diminishing andmaximum-principle-preserving with {at least twice} larger time step constraintsthan the classical first order Eulerian method for Burgers' equation. Numericalexperiments verify the optimality of the designed time step sizes.

Stability analysis of the Eulerian-Lagrangian finite volume methods for nonlinear hyperbolic equations in one space dimension