  In this paper we present and analyze a general framework for constructing
high order explicit local time stepping (LTS) methods for hyperbolic
conservation laws. In particular, we consider the model problem discretized by
Runge-Kutta discontinuous Galerkin (RKDG) methods and design LTS algorithms
based on strong stability preserving Runge-Kutta (SSP-RK) schemes, that allow
spatially variable time step sizes to be used for time integrations in
different regions. The proposed algorithms are of predictor-corrector type, in
which the interface information along the time direction is first predicted
based on the SSP-RK approximations and Taylor expansions, and then the fluxes
over the region of interface are corrected to conserve mass exactly at each
time step. Following the proposed framework, we detail the corresponding LTS
schemes with accuracy up to the fourth order, and prove their conservation
property and nonlinear stability for the scalar conservation laws. Numerical
experiments are also presented to demonstrate excellent performance of the
proposed LTS algorithms.
