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Register a new userA Local Discontinuous Galerkin Level Set Reinitialization with Subcell Stabilization on Unstructured Meshes
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Added by
Ali Karakus
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this paper we consider a level set reinitialization technique based on a high-order, local discontinuous Galerkin method on unstructured triangular meshes. A finite volume based subcell stabilization is used to improve the nonlinear stability of the method. Instead of the standard hyperbolic level set reinitialization, the flow of time Eikonal equation is discretized to construct an approximate signed distance function. Using the Eikonal equation removes the regularization parameter in the standard approach which allows more predictable behavior and faster convergence speeds around the interface. This makes our approach very efficient especially for banded level set formulations. A set of numerical experiments including both smooth and non-smooth interfaces indicate that the method experimentally achieves design order accuracy.
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In this paper, we generalize the compact subcell weighted essentially non oscillatory (CSWENO) limiting strategy for Runge-Kutta discontinuous Galerkin method developed recently by us in 2021 for structured meshes to unstructured triangular meshes. The main idea of the limiting strategy is to divide the immediate neighbors of a given cell into the required stencil and to use a WENO reconstruction for limiting. This strategy can be applied for any type of WENO reconstruction. We have used the WENO reconstruction proposed by Zhu and Shu in 2019 and provided accuracy tests and results for two-dimensional Burgers equation and two dimensional Euler equations to illustrate the performance of this limiting strategy.
Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.
A priori subcell limiting approach is developed for high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) on two-dimensional unstructured quadrilateral meshes. Firstly, a modified indicator based on modal energy coefficients is proposed to detect troubled cells. Then, troubled cells are decomposed into nonuniform subcells and each subcell has one solution point. A second-order finite difference shock-capturing scheme based on nonuniform nonlinear weighted (NNW) interpolation is constructed to calculate troubled cells while smooth cells are calculated by the CPR method. Numerical investigations show that the subcell limiting strategy on unstructured quadrilateral meshes is robust in shock-capturing.
We propose a Discontinuous Galerkin method for the Poisson equation on polygonal tessellations in two dimensions, stabilized by penalizing, locally in each element $K$, a residual term involving the fluxes, measured in the norm of the dual of $H^1(K)$. The scalar product corresponding to such a norm is numerically realized via the introduction of a (minimal) auxiliary space inspired by the Virtual Element Method. Stability and optimal error estimates in the broken $H^1$ norm are proven under a weak shape regularity assumption allowing the presence of very small edges. The results of numerical tests confirm the theoretical estimates.
We introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree $p$. In the setting of [S. Bertoluzza and D. Prada, A polygonal discontinuous Galerkin method with minus one stabilization, ESAIM Math. Mod. Numer. Anal. (DOI: 10.1051/m2an/2020059)], the stabilization is obtained by penalizing, in each mesh element $K$, a residual in the norm of the dual of $H^1(K)$. This negative norm is algebraically realized via the introduction of new auxiliary spaces. We carry out a $p$-explicit stability and error analysis, proving $p$-robustness of the overall method. The theoretical findings are demonstrated in a series of numerical experiments.
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