Multiwavelets and outlier detection for troubled-cell indication in discontinuous Galerkin methods

This dissertation addresses practical use of multiwavelets and outlier detection for troubled-cell indication for discontinuous Galerkin (DG) methods. For smooth solutions, the DG approximation converges to the exact solution with a high order of accuracy. However, problems may arise when shock waves or discontinuities appear: non-physical spurious oscillations are formed close to these discontinuous regions. These oscillations can be prevented by applying a limiter near these regions. One of the difficulties in using a limiter is identifying the difference between a true discontinuity and a local extremum of the approximation. Troubled-cell indicators can help to detect this difference and identify the discontinuous regions (so-called ’troubled cells’) where a limiter should be applied.
In this dissertation, a multiwavelet formulation is used to decompose the DG approximation. The multiwavelet coefficients act as a troubled-cell indicator since they suddenly increase in the neighborhood of a discontinuity. This leads to the definition of a new multiwavelet indicator that detects elements as troubled if the coefficient is large enough in absolute value. Here, a problem-dependent parameter is needed to define the strictness of the indicator. To forgo the reliance on a parameter, a new outlier-detection algorithm is defined that uses boxplot theory. This method can also be applied to different troubled-cell indicators.
Results are shown for regular one-dimensional and tensor-product two-dimensional meshes, as well as for irregular meshes in one dimension and triangular meshes in two dimensions.