High order implicit time integration schemes on multiresolution adaptive   grids for stiff PDEs
  We consider high order, implicit Runge-Kutta schemes to solve time-dependent stiff PDEs on dynamically adapted grids generated by multiresolution analysis for unsteady problems disclosing localized fronts. The multiresolution finite volume scheme yields highly compressed representations within a user-defined accuracy tolerance, hence strong reductions of computational requirements to solve large, coupled nonlinear systems of equations. SDIRK and RadauIIA Runge-Kutta schemes are implemented with particular interest in those with L-stability properties and accuracy-based time-stepping capabilities. Numerical evidence is provided of the computational efficiency of the numerical strategy to cope with highly unsteady problems modeling various physical scenarios with a broad spectrum of time and space scales. 
Max Duarte, Richard Dobbins, Mitchell Smooke
math.NA math.AP