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

In order to accelerate finite-volume schemes (FVSs) applied to (inhomogeneous) hyperbolic conservation laws multiresolution-based adaptive concepts can be used. The basic idea is to analyse the local regularity by means of a multiresolution analysis of cell averages. By difference information between successive refinement levels, local grid adaptation is triggered employing threshold techniques. This leads to a significant gain in computational complexity. The crucial point is to compute numerical fluxes and sources on local resolution levels such that the perturbation error introduced by the adaptive procedure due to thresholding is of the same order as the discretization error of the reference FVS on the finest uniform discretization. In the present work a modified approach based on polynomial reconstruction techniques is introduced and investigated analytically. The efficiency and accuracy of the adaptive concept is significantly improved in comparison to previous approaches, in particular, for inhomogeneous equations. This is confirmed by numerical parameter studies.