An efficient local time-stepping scheme for solution of nonlinear conservation laws

We develop an efficient local time-stepping algorithm for the method of lines approach to numerical solution of transient partial differential equations. The need for local time-stepping arises when adaptive mesh refinement results in a mesh containing cells of greatly different sizes. The global CFL number and, hence, the global time step, are defined by the smallest cell size. This can be inefficient as a few small cells may impose a restrictive time step on the whole mesh. A local time-stepping scheme allows us to use the local CFL number which reduces the total number of function evaluations. The algorithm is based on a second order Runge–Kutta time integration. Its important features are a small stencil and the second order accuracy in the L2 and L∞ norms.