A Multi-wavelet type limiter for discontinuous Galerkin approximations

In this report, we present a multi-wavelet type limiter for the discontinuous Galerkin method for limiting the solution when spurious oscillations develop near a shock. This limiting leads to a loss of information in the approximation that can be detrimental to a higher order approximation (k > 2). The goal is therefore to retain as much information as possible in the higher order approximation. This is done by taking advantage of the evolution in time of more degrees of freedom of a DG approximation by making use of ideas from multi-resolution analysis (MRA) [3]. This differs from multi-level method [18] in that it only seeks to apply MRA ideas locally, on elements where the approximation requires limiting. This combination of techniques seems a natural pairing as it is well known that the wavelet linear approximation (i.e., truncating the high frequencies) can approximate smooth functions very efficiently. Previously, the major hurdle was in devising wavelets that satisfy boundary conditions. With the discontinuous Galerkin method this is no longer an issue. Multi-wavelets can achieve arbitrary high accuracy without Gibbs’ phenomena by selecting an appropriate wavelet basis, concentrating the energy to low frequencies. Standard wavelet linear approximation techniques cannot achieve similar results for functions which are not smooth, such as piecewise continuous functions with large jumps. In this paper we present results showing that the multi-wavelet idea is a promising technique for limiting solutions