https://www.brown.edu/research/projects/scientific-computing/sites/brown.edu.research.projects.scientific-computing/files/uploads/High%20order%20WENO%20and%20DG%20methods%20for%20time-dependent%20convection-dominated%20PDEs.pdf

High order WENO and DG methods for time-dependent convection-dominated PDEs

On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier-Stokes equations

This paper proposes an explicit, (at least) second-order, maximum principle sat- isfying, Lagrange finite element method for solving nonlinear scalar conservation equations. The technique is based on a new viscous bilinear form introduced in Guermond and Nazarov (Com- put. Methods Appl. Mech. Engrg., 272 (2014), pp. 198-213), a high-order entropy viscosity method, and the Boris-Book-Zalesak flux correction technique. The algorithm works for arbitrary meshes in any space dimension and for all Lipschitz fluxes. The formal second-order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems.

/pdf/a-second-order-maximum-principle-preserving-lagrange-finite-3pzae7sclo.pdf

A second-order maximum principle preserving lagrange finite element technique for nonlinear scalar conservation equations ∗

Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes

https://www.math.tamu.edu/~guermond/PUBLICATIONS/guermond_nazarov_cmame_2014.pdf

A maximum-principle preserving finite element method for scalar conservation equations

https://www.brown.edu/research/projects/scientific-computing/sites/brown.edu.research.projects.scientific-computing/files/uploads/Maximum-principle.pdf

Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-diffusion equations on triangular meshes

In this paper, we propose a multi-scale discontinuous Galerkin (DG) method for second-order elliptic problems with curvilinear unidirectional rough coefficients by choosing a special non-polynomial approximation space. The key ingredient of the method lies in the incorporation of the local oscillatory features of the differential operators into the approximation space so as to capture the multi-scale solutions without having to resolve the finest scales. The unidirectional feature of the rough coefficients allows us to construct the basis functions of the DG non-polynomial approximation space explicitly, thereby greatly increasing the algorithm efficiency. Detailed error estimates for two-dimensional second-order DG methods are derived, and a general guidance on how to construct such non-polynomial basis is discussed. Numerical examples are also presented to validate and demonstrate the effectiveness of the algorithm.

/pdf/multi-scale-discontinuous-galerkin-method-for-solving-54u39v9lnl.pdf

Yifan Zhang

Papers

Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection-diffusion equations on triangular meshes

Multi-scale Discontinuous Galerkin Method for Solving Elliptic Problems with Curvilinear Unidirectional Rough Coefficients