Adaptive multiresolution discontinuous Galerkin schemes for conservation laws

Abstract

A multiresolution-based adaptation concept is proposed that aims at accelerating discontinuous Galerkin schemes applied to non-linear hyperbolic conservation laws. Opposite to standard adaptation concepts no error estimates are needed to tag mesh elements for refinement. Instead of this, a multiresolution analysis is performed on a hierarchy of nested grids for the data given on a uniformly refined mesh. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. A central mathematical problem addressed in this work is then to show at least for scalar one-dimensional problems that choosing an appropriate threshold value, the adaptive solution is of the same accuracy as the reference solution on a uniformly refined mesh. Numerical comparisons demonstrate the efficiency of the concept.

Study the Performance of the Two Wavelet-Based Adaptation Schemes for the Shallow Water Flow Modelling

“…[12]). These techniques rely on the a priori or posterior knowledge of the numerical solution to produce an adaptive mesh and this leads to introduce uncertainties into the numerical solution [13][14] and to be compromised an error-sensor is designed within the numerical scheme. However, an accurate mathematical approach is not available yet to determine an accurate error-sensor for the system of SWE [13].…”

“…These techniques rely on the a priori or posterior knowledge of the numerical solution to produce an adaptive mesh and this leads to introduce uncertainties into the numerical solution [13][14] and to be compromised an error-sensor is designed within the numerical scheme. However, an accurate mathematical approach is not available yet to determine an accurate error-sensor for the system of SWE [13]. To avoid using these types of error-estimator sensors the theory of discretewavelets theory incorporates with the context of finite volume method and its extension to higher order which is called discontinues Galerkin method are used for solving the system of conservation law (see e.g.…”

Multiwavelet-based grid adaptation with discontinuous Galerkin schemes for shallow water equations

“…In recent years the concept was extended to the framework of DG schemes. Originally in [49,50] the concept was analytically and numerically investigated for the one-dimensional scalar case. The method was then extended to the Euler equations [51] and later on to the multi-dimensional case [52].…”

Automated Parameters for Troubled-Cell Indicators Using Outlier Detection

“…The choice for half-open intervals follows from the paper of Archibald, Fann, and Shelton [3]. Different choices are available in the literature, for example, closed intervals (Hovhannisyan, Müller, and Schäfer [17]), or open intervals (Gerhard et al [11]). …”

“…However, when the solution is sufficiently smooth, then the element-boundary jumps in the approximation and its derivatives will be noticeably smaller than when a discontinuity in Downloaded 01/27/16 to 130.161.210.53. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php (one of the derivatives of) the solution is present due to the cancellation property of multiwavelets [18]. The multiwavelet coefficientsd n−1 kj are used to detect troubled cells when…”