Discontinuous Galerkin Method with an Entropic Slope Limiter for Euler Equations
- PDF / 564,824 Bytes
- 10 Pages / 612 x 792 pts (letter) Page_size
- 8 Downloads / 302 Views
ontinuous Galerkin Method with an Entropic Slope Limiter for Euler Equations M. D. Bragina, b, *, Y. A. Kriksina, **, and V. F. Tishkina, *** a
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, 125047 Russia b Moscow Institute of Physics and Technology, Dolgoprudny, 141701 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected] Received June 17, 2019; revised June 17, 2019; accepted September 9, 2019
Abstract—The variational approach to obtaining equations of the entropy stable discontinuous Galerkin method is generalized. It is shown how the monotonicity property can be incorporated into this approach. As applied to Euler equations, the entropic slope limiter, a new effective approximate method for the studied approach, is designed. It guarantees the monotonicity of the numerical solution, as well as the nonnegativity of the pressure and entropy production for each finite element. This method is successfully tested on some well-known gas dynamics model problems. Keywords: gas dynamics equations, discontinuous Galerkin method, slope limiter, entropy inequality DOI: 10.1134/S2070048220050038
1. INTRODUCTION Lately, entropy stable methods and algorithms for resolving gas dynamics problems and problems of similar types are being actively developed (see [1–10]). This is being done to improve the quality of the numerical solutions, including in the numerical algorithms, in addition to the conservation laws traditionally taken into account, the second law of thermodynamics quantitatively expressed by the entropy inequality. The second law of thermodynamics admits not only irreversible processes but also reversible processes (in this case, the inequality turns into an equality); however, it does not allow decreasing the entropy in a continuous medium (see [11]). These methods include the discontinuous Galerkin method, which proved to be highly efficient in problems of continuum mechanics (see [10, 12–18]). A detailed review of papers related to the entropy stability of numerical methods can be found in [19]. In [20], a method of variational entropic regularization is proposed, where equations of the discontinuous Galerkin method are defined through the solution of the quadratic programming problem in which the second power of the residual norm for the classical equations of the discontinuous Galerkin method (for Euler equations) is conditionally minimized over the set defined by a linear inequality that is a discrete analog of the entropy condition. However, according to the numerical testing (see [19]), such an entropic regularization does not generate a stable computational algorithm. In problems with shock waves and rarefaction waves propagating in the opposite directions, the specified methods do not prevent the appearance of negative pressure values within a finite time regardless of the smallness of the time step. Thus, the fulfillment of discrete analogs of the conservation laws and the entropy inequality does not guarantee the workability of algori
Data Loading...
