Numerical Approximations to Nonlinear Conservation Laws with Locally Varying Time and Space Grids

An explicit time differencing technique is introduced to approximate nonlinear conservation laws. This differencing technique links together an arbitrary number of space regimes containing fine and coarse time increments. Previous stability requirements, i.e. the CFL condition, placed a global bound on the size of the time increments. For scalar, monotone, approximations in one space dimension, using this variable step time differencing, convergence to the correct physical solution is proven given only a local CFL condition.