Positivity Preserving Limiters for Time-Implicit Higher Order Accurate Discontinuous Galerkin Discretizations

Positivity-Preserving Discontinuous Galerkin Methods with Lax-Wendroff Time Discretizations

This work introduces a single-stage, single-step method for the compressible Euler equations that is provably positivitypreserving and can be applied on both Cartesian and unstructured meshes. This method is the first case of a singlestage, single-step method that is simultaneously high-order, positivity-preserving, and operates on unstructured meshes. Time-stepping is accomplished via the Lax-...

Domain Decomposition Preconditioners for Higher-Order Discontinuous Galerkin Discretizations

Aerodynamic flows involve features with a wide range of spatial and temporal scales which need to be resolved in order to accurately predict desired engineering quantities. While computational fluid dynamics (CFD) has advanced considerably in the past 30 years, the desire to perform more complex, higher-fidelity simulations remains. Present day CFD simulations are limited by the lack of an effi...

Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations

Efficient solution techniques for high-order accurate time-dependent problems are investigated for solving the two-dimensional non-linear Euler equations in this work. The spatial discretization consists of a high-order accurate Discontinuous Galerkin (DG) approach. Implicit time-integration techniques are considered exclusively in order to avoid the stability restrictions of explicit methods. ...

Implicit Positivity-Preserving High-Order Discontinuous Galerkin Methods for Conservation Laws

Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge-Kutta time discretizations

This paper investigates the use of a special class of strong-stability-preserving (SSP) Runge–Kutta time discretization methods in conjunction with discontinuous Galerkin (DG) finite element spatial discretizatons. The class of SSP methods investigated here is defined by the property that the number of stages s is greater than the order k of the method. From analysis, CFL conditions for the lin...