Wavelets and adaptive grids for the discontinuous Galerkin method (original) (raw)
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Aspects of discontinuous Galerkin methods for hyperbolic conservation laws
Finite Elements in …, 2002
We review several properties of the discontinuous Galerkin method for solving hyperbolic systems of conservation laws including basis construction, ux evaluation, solution limiting, adaptivity, and a posteriori error estimation. Regarding error estimation, we show that the leading term of the spatial discretization error using the discontinuous Galerkin method with degree p piecewise polynomials is proportional to a linear combination of orthogonal polynomials on each element of degrees p and p+1. These are Radau polynomials in one dimension. The discretization errors have a stronger superconvergence of order O(h 2p+1 ), where h is a mesh-spacing parameter, at the out ow boundary of each element. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors in regions where solutions are smooth.
Discontinuous Galerkin methods for the solution of a class of hyperbolic problems
A numerical solution of one dimensional (1D) system of conservation laws is presented based on the Runge Kutta Discontinuous Galerkin (RKDG) method. We present and analyse the RKDG method and implement it for the Euler equations of gas dynamics and the shallow water equations for various initial data. The RKDG method is a very attractive method because of its formal high-order accuracy, its ability to handle complicated geometries, and its ability to capture discontinuities without producing spurious oscillations. Declaration I, the undersigned, hereby declare that the work contained in this research project is my original work, and that any work done by others or by myself previously has been acknowledged and referenced accordingly.
A parallel hp-adaptive discontinuous Galerkin method for hyperbolic conservation laws
Applied Numerical Mathematics, 1996
This paper describes a parallel algorithm based on discontinuous hp-nite element approximations of linear, scalar, hyperbolic conservation laws. The paper focuses on the development of an e ective parallel adaptive strategy for such problems. Numerical experiments suggest that these techniques are highly parallelizable and exponentially convergent, thereby yielding e ciency many times superior to conventional schemes for hyperbolic problems.
Stabilized discontinuous Galerkin method for hyperbolic equations
Computer Methods in Applied Mechanics and Engineering, 2005
In this work a new stabilization technique is proposed and studied for the discontinuous Galerkin method applied to hyperbolic equations. In order to avoid the use of slope limiters, a streamline diffusion-like term is added to control oscillations for arbitrary element orders. Thus, the scheme combines ideas from both the Runge-Kutta discontinuous Galerkin method [J. Scient. Comput. 16 (2001) 173] and the streamline diffusion method [Comput. Methods Appl. Mech. Engrg. 32 (1982)]. To increase the stability range of the method, the diffusion term is treated implicitly. The result is a scheme with higher order in space with the same stability range as the finite volume method. An optimal relation between the time step and the size of the diffusion coefficient is analyzed for numerical precision. The scheme is implemented using the object oriented programming philosophy based on the environment described in [Comput. Methods Appl. Mech. Engrg. 150 (1997)]. Accuracy and shock capturing abilities of the method are analyzed in terms of two bidimensional model problems: the rotating cone and the backward facing step problem for the Euler equations of gas dynamics.
hp-Version discontinuous Galerkin methods for hyperbolic conservation laws
Computer Methods in Applied Mechanics and Engineering, 1996
The development of hp-version discontinuous Galerkin methods for hyperbolic conservation laws is presented in this work. A priori error estimates are derived for a model class of linear hyperbolic conservation laws. These estimates are obtained using a new mesh-dependent norm that reflects the dependence of the approximate solution on the local element size and the local order of approximation.~The results generalize and extend previous results on meshdependent norms to hp-version discontinuous Galerkin methods. A posteriori error estimates which provide bounds on the actual error are also developed in this work. Numerical experiments verify the a priori estimates and demonstrate the effectiveness of the a posteriori estimates in providing reliable estimates of the actual error in the numerical solution.
An adaptive wavelet viscosity method for hyperbolic conservation laws
Numerical Methods for Partial Differential Equations, 2008
We extend the multiscale finite element viscosity method for hyperbolic conservation laws developed in terms of hierarchical finite element bases to a (pre-orthogonal spline-)wavelet basis. Depending on an appropriate error criterion, the multiscale framework allows for a controlled adaptive resolution of discontinuities of the solution. The nonlinearity in the weak form is treated by solving a least-squares data fitting problem.
Computational and Applied Mathematics, 2014
The concept of multiresolution-based adaptive DG schemes for nonlinear one-dimensional hyperbolic conservation laws has been developed and investigated analytically and numerically in N. Hovhannisyan, S. Müller, R. Schäfer, Adaptive multiresolution Discontinuous Galerkin Schemes for Conservation Laws, Math. Comp., 2013. The key idea is to perform a multiresolution analysis using multiwavelets 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. The focus of the present work lies on the extension of the originally one-dimensional concept to higher dimensions and the verification of the choice for the threshold value by means of parameter studies performed for linear and nonlinear scalar conservation laws.
International Journal for Numerical Methods in Engineering, 1995
This paper describes a parallel algorithm based on discontinuous hp-finite element approximations of linear, scalar, hyperbolic conservation laws. The paper focuses on the development of an effective parallel adaptive strategy for such problems. Numerical experiments suggest that these techniques are highly parallelizable and exponentially convergent, thereby yielding efficiency many times superior to conventional schemes for hyperbolic problems.
An adaptive wavelet space‐time SUPG method for hyperbolic conservation laws
Numerical Methods for Partial Differential Equations, 2017
This article concerns with incorporating wavelet bases into existing streamline upwind Petrov‐Galerkin (SUPG) methods for the numerical solution of nonlinear hyperbolic conservation laws which are known to develop shock solutions. Here, we utilize an SUPG formulation using continuous Galerkin in space and discontinuous Galerkin in time. The main motivation for such a combination is that these methods have good stability properties thanks to adding diffusion in the direction of streamlines. But they are more expensive than explicit semidiscrete methods as they have to use space‐time formulations. Using wavelet bases we maintain the stability properties of SUPG methods while we reduce the cost of these methods significantly through natural adaptivity of wavelet expansions. In addition, wavelet bases have a hierarchical structure. We use this property to numerically investigate the hierarchical addition of an artificial diffusion for further stabilization in spirit of spectral diffusio...
Adaptive multiresolution discontinuous Galerkin schemes for conservation laws
Mathematics of Computation, 2013
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.