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.
Dissertation