A Hybridized Discontinuous Petrov-Galerkin Method for Compresible Flows

Abstract

We present a hybridized discontinuous Petrov-Galerkin (HDPG) method for the solution of compressible flows. The HPDG method combines the efficiency of the hybridizable discontinuous Galerkin (HDG) method with the excellent stability of the discontinuous Petrov-Galerkin (DPG) method. This aim is achieved by using the DGP method to discretize the governing equations at the element level and the HDG method to glue the local solutions together. Moreover, we propose to enrich the test space with a constant function in order to make the HDPG method conservative. In the presence of under-resolved features such as shocks and boundary layers, the HDPG scheme is found to be more robust and stable than the HDG method. We present several numerical examples to demonstrate the proposed method. The development of robust, accurate, and efficient methods for the numerical solution of hyperbolic systems of conservation laws in complex geometries is a topic of considerable importance. Indeed, hyperbolic systems of conservation laws govern a wide range of physical phenomena and arise in several areas of applied mathematics and mechanics such as fluid dynamics, thermodynamics, population dynamics, magnetohydrodynamics, multiphase flow in nonlinear material, and traffic flow. The most fundamental phenomenon of hyperbolic systems is the formation and propagation of discontinuities and shock waves even if initial and boundary data are smooth. The presence of shock waves is a serious challenge for any numerical methods to provide a physical and stable solution. The main difficulties in computing solutions with shocks are that (1) when a shock is formed it poses a source of instability in the shock region, which then leads to numerical instabilities if no treatment of shock waves is introduced; (2) it is hard to predict when and where new shocks arise, and track them as they propagate; (3) solution must satisfy the Rankine-Hugoniot jump condition and the entropy conditions; (4) solution should have sharp and clean shocks at the discontinuity interface; and (5) numerical treatment of shock waves should not cause deterioration in resolution and reduction of accuracy in domains where the solution is smooth. Although significant progress has been made over the years in both the theoretical and numerical investigations, capturing shocks, especially when shocks propagate and interact with one another, remains an active research area with many challenging problems to be addressed. In recent years, considerable attention has been turned to discontinuous Galerkin (DG) methods 1–4,8,9,11,14–17,30,32 for the numerical solution of hyperbolic systems of conservation laws. DG methods possess several attractive properties for solving hyperbolic problems. In particular, they are flexible for complicated geometry, locally conservative, high-order accurate, highly parallelizable, and have low dissipation and dispersion. However, in spite of all these advantages, DG methods have not yet made a more significant impact for practical applications. This is largely due to the main criticism that DG methods are computationally expensive. This cost is primarily associated to the large number of degrees of freedom caused by nodal duplication at the element boundary interfaces. More specifically, assuming about six linear tetrahedral elements per node, the number of unknowns in a DG system would approximately be 24 times the number of unknowns in the corresponding continuous Galerkin (CG) system for the same order of the approximating polynomial. The storage and computation cost of implicit DG methods are thus several times that of CG methods. However,