Detecting Discontinuities Over Triangular Meshes Using Multiwavelets

It is well known that solutions to nonlinear hyperbolic PDEs develop discontinuities in time. The generation of spurious oscillations in such regions can be prevented by applying a limiter in the troubled zones. In earlier work, we constructed a multiwavelet troubled-cell indicator for one and (tensor-product) two dimensions (SIAM J. Sci. Comput. 38(1):A84–A104, 2016). In this paper, we investigate multiwavelet troubled-cell indicators on structured triangular meshes. One indicator uses a problem-dependent parameter; the other indicator is combined with outlier detection

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.Mathematical Physic

Reconstructie van de aardkorst met tomografie (Reconstruction of the earth’s crust by tomography)

Röntgenfoto’s, echo’s en reconstructies van de aardkorst zijn drie toepassingen waarbij tomografie een grote rol speelt. Tomografie is een methode om eigenschappen van een onbereikbaar gebied te bepalen met behulp van stralen of drukgolven. De aardkorst bestaat uit allerlei verschillende steenlagen. Als op grote diepte een aardbeving plaatsvindt, dan hebben verschillende drukgolven die vertrekken vanaf het epicentrum verschillende reistijden naar het aardoppervlak. Deze meetbare reistijden hangen onder andere af van de voortplantingssnelheid die iedere steensoort heeft. Bij dit onderzoek is een model gebruikt om golfvoortplantingssnelheden te bepalen aan de hand van reistijdmetingen. Met behulp van dit model is het mogelijk de aardkorst op grote diepte (tientallen kilometers) te reconstrueren. Tomografie is dus een krachtig hulpmiddel om aardbevingen te kunnen voorspellen. Omdat er altijd meetfouten zullen optreden, zal de gevonden oplossing afwijken van de echte oplossing. Drie numerieke methoden die de verstoring door meetfouten onderdrukken, zijn singuliere waardenontbinding, Tikhonov regularisatie en ART. In het verslag wordt de werking van deze methoden onderzocht en vergeleken. Hierbij draait het niet alleen om de kwaliteit van het gezochte plaatje, maar ook om de hoeveelheid operaties die gebruikt wordt en het geheugengebruik dat nodig is.Numerical AnalysisApplied mathematicsElectrical Engineering, Mathematics and Computer Scienc

Limiting and shock detection for discontinuous Galerkin solutions using multiwavelets

Many areas such as climate modeling, shallow water equations, and computational fluid dynamics use nonlinear hyperbolic partial differential equations (PDE’s) to describe the behaviour of some unknown quantity. In general, the solutions of these equations contain shocks, or develop discontinuities. To solve these equations, various types of numerical methods can be used, such as finite difference, finite volume and finite element methods. In this master thesis, the discontinuous Galerkin method (DG) is used. To efficiently apply DG in case of discontinuous solutions, limiting techniques are used to reduce the spurious oscillations, that are developed in the discontinuous regions. Unfortunately, most of the limiters do not work well for higher order approximations, or multidimensional cases. Originally, the project focused on limiting DG solutions using multiwavelets. However, it is very hard to limit the solution using the multiwavelet decomposition. We did discover that the multiwavelet expansion is quite practical for shock detection. This report describes the pros and cons of multiwavelets as a limiter and shock detector for limiting DG.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc