An adaptive order Godunov type central scheme

Traditionally, high order Godunov-type central schemes employ local polynomial reconstructions. These reconstructions avoid transfer of information across discontinuities through nonlinear limiters which act as local edge detectors. Here we introduce an adaptive method which employ global edge detection to ensure that information is extracted in the direction of smoothness while maintaining computational stability. Additionally, the global edge detection substantially reduces the computational cost. The reconstruction incorporates the largest symmetric stencil possible without crossing discontinuities. Consequently, the spatial order of accuracy is proportional to the number of cells to the nearest discontinuity, reaching exponential order at the interior of regions of smoothness.