On the stability of fully adaptive multiscale schemes for conservation laws using approximate flux and source reconstruction strategies

Adaptive Multiscale Methods for Flow Problems: Recent Developments

“…In the present work we explain how to integrate the multiresolution anaysis in the flow solver by presenting recent developments on (i) a local strategy to compute numerical fluxes and sources on locally refined grids with hanging nodes, cf. [17], (ii) a multilevel time stepping strategy where refinement is also performed in time, cf. [23], and (iii) an FAS-like strategy to solve efficiently the nonlinear problems arising from an implicit time discretization of the underlying finite volume scheme, cf.…”

“…We refer to (17) and (18) as exact flux and source reconstruction, respectively. Since in (18) we have to compute all sources on the finest scale, there is no complexity reduction, i.e., we still have the complexity of the reference grid.…”

“…(Evolution) Evolve the cell averages associated to the pre-refined gridG n+1 L,ε according to (15) where the numerical fluxes and sources are not necessarily determined by (17) and (18), respectively, i.e.,…”

“…An ideal choice would maintain the accuracy of the reference scheme at reduced computational cost. For a detailed analysis we refer to [11,17] and explain here only the main ideas..…”

“…de are not just treated as independent black boxes communicating via interfaces but are highly intertwined on a conceptual level mainly linking (i) the multiresolutionbased grid adaption that reliably detects and resolves all physical relevant effects, and (ii) the B-spline grid generator which reduces grid changes to just moving a "few" control points whose number is, in particular, independent of any local grid refinements. Over the past few years the above framework has been further enriched and extended by mathematical concepts, such as multilevel time stepping [20,19,23], multilevel solver strategies [24], time step control [29,27,28], adaptive local flux and source reconstruction [17], as well as techniques from computer science. In particular, parallelization of the multiresolution grid adaptation using space-filling curves [31], has been incorporated to further improve the efficiency of Quadflow.…”