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.
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.
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.
A series of shock capturing schemes based on nonuniform nonlinear weighted interpolation on nonuniform points are developed for conservation laws. Smoothness indicator and discrete conservation laws are discussed. To make fair comparisons between different types of schemes, the properties of eigenvalues of spatial discretization matrices are proved. And the proposed schemes are compared with Weighted Compact Nonlinear Schemes (WCNS) and Flux Reconstruction or Correction Procedure via Reconstruction (FR/CPR) in dispersion, dissipation properties and numerical accuracy. Then, the proposed shock capturing schemes are used as subcell limiters for high-order FR/CPR and the hybrid scheme has superiority in data transformation and satisfying discrete conservation laws. Accuracy, discrete conservation laws and shock capturing properties are tested. Numerical results in one and two dimensions are provided to illustrate that the proposed schemes have good properties in shock capturing and can be applied as subcell limiters for FR/CPR.
In this work we propose a discretisation method for the Reissner--Mindlin plate bending problem in primitive variables that supports general polygonal meshes and arbitrary order. The method is inspired by a two-dimensional discrete de Rham complex for which key commutation properties hold that enable the cancellation of the contribution to the error linked to the enforcement of the Kirchhoff constraint. Denoting by $kge 0$ the polynomial degree for the discrete spaces and by $h$ the meshsize, we derive for the proposed method an error estimate in $h^{k+1}$ for general $k$, as well as a locking-free error estimate for the lowest-order case $k=0$. The theoretical results are validated on a complete panel of numerical tests.
In this work, we develop a discretisation method for the mixed formulation of the magnetostatic problem supporting arbitrary orders and polyhedral meshes. The method is based on a global discrete de Rham (DDR) sequence, obtained by patching the local spaces constructed in [Di Pietro, Droniou, Rapetti, Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra, arXiv:1911.03616] by enforcing the single-valuedness of the components attached to the boundary of each element. The first main contribution of this paper is a proof of exactness relations for this global DDR sequence, obtained leveraging the exactness of the corresponding local sequence and a topological assembly of the mesh valid for domains that do not enclose any void. The second main contribution is the formulation and well-posedness analysis of the method, which includes the proof of uniform Poincare inequalities for the discrete divergence and curl operators. The convergence rate in the natural energy norm is numerically evaluated on standard and polyhedral meshes. When the DDR sequence of degree $kge 0$ is used, the error converges as $h^{k+1}$, with $h$ denoting the meshsize.