An adaptive multiresolution discontinuous Galerkin scheme for conservation laws

• Ein adaptives multiskalen unstetiges Galerkin Verfahren für Erhaltungsgleichungen

Gerhard, Nils; Müller, Siegfried (Thesis advisor); Dahmen, Wolfgang (Thesis advisor)

Aachen (2017)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017

Abstract

This thesis focuses on the development of adaptive multiresolution-based discontinuous Galerkin schemes for conservation laws. The key idea of this concept is to accelerate a given reference scheme on a uniformly refined grid by computing actually only on a locally adapted subgrid while maintaining its accuracy. The dynamical grid adaptation is based on a multiresolution analysis of the underlying data, where the data corresponding to the current solution are represented as data on some coarse level and difference information called details corresponding to successive refinement levels. By local thresholding of small details the adaptive grid is determined. In previous work by N. Hovhannisyan, S. Müller and R. Schäfer (Math. Comp., 2014) the basic concept has been introduced and investigated for one-dimensional scalar problems using uniform dyadic grid hierarchies. This setting significantly restricts the range of application. To overcome this limitation the present thesis aims at (i) developing a wavelet-free multiresolution analysis to deal with non-uniform grid hierarchies, (ii) deriving a reliable and efficient strategy to choose the threshold parameters, (iii) providing a strategy to deal with constraints of the underlying partial differential equations and (iv) validating the concept for non-linear systems of conservation laws in multiple space dimensions. A crucial part in the realization of the adaptive scheme is the construction of generators for the details in difference spaces. The construction of a basis for these spaces is computationally costly. To overcome this issue an alternative approach to realize the grid adaptation is developed which does not rely on the construction of these bases functions and enables a straightforward application of the adaptive strategy to arbitrary nested grid hierarchies. The efficiency and reliability of the adaptive scheme is strongly influenced by the choice of the local threshold values. For that purpose, a robust and reliable strategy is developed for choosing the local threshold values without any tuning, parameter fitting or need of computing a solution on the (uniform) reference grid. Many problems are subjected to constraints. Exemplarily, constraints in the context of the shallow water equations are considered. In particular, a discontinuous Galerkin scheme for the shallow water equations must be able to preserve steady states over non-constant topography as well as positive values of depth. Local grid adaptation might spoil these discrete constraints. For that purpose, a strategy to deal with these constraints is developed and it is proven that they are maintained during grid adaption. For the purpose of validating the concept, adaptive computations have been performed for several well-known benchmark test cases for the Burgers' equation, the compressible Euler equations, the shallow water equations as well as the compressible Navier-Stokes equations. These show that the adaptive scheme is capable of accelerating the reference scheme significantly while maintaining its accuracy.