Implicit shock tracking using an optimization-based high-order discontinuous Galerkin method

JOURNAL OF COMPUTATIONAL PHYSICS

卷:410

Article

Zahr, M. J.¹  Shi, A.²  Persson, P. -O^2,3 

[1] Univ Notre Dame, Dept Aerosp & Mech Engn, Notre Dame, IN 46556 USA

[2] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA

[3] Lawrence Berkeley Natl Lab, Math Grp, 1 Cyclotron Rd, Berkeley, CA 94720 USA

关键词: Shock tracking;    Shock fitting;    r-adaptivity;    High-order methods;    Discontinuous Galerkin;    High-speed flows;   

DOI  :  10.1016/j.jcp.2020.109385

来源: Elsevier

【 摘 要 】

A novel framework for resolving discontinuous solutions of conservation laws, e.g., contact lines, shock waves, and interfaces, using implicit trackingand a high-order discontinuous Galerkin (DG) discretization was introduced in [39]. Central to the framework is an optimization problem whose solution is a discontinuity-aligned mesh and the corresponding high-order approximation to the flow that does not require explicit meshing of the unknown discontinuity surface. The method was shown to deliver highly accurate solutions on coarse, high-order discretizations without nonlinear stabilization and recover optimal convergence rates O(h(p+1)) even for problems with discontinuous solutions. This work extends the implicit tracking framework such that robustness is improved and convergence accelerated. In particular, we introduce an improved formulation of the central optimization problem and an associated sequential quadratic programming (SQP) solver. The new error-based objective function penalizes violation of the DG residual in an enriched test spaceand is shown to have excellent tracking properties. The SQP solver simultaneously converges the nodal coordinates of the mesh and DG solution to their optimal values and is equipped with a number of features to ensure robust, fast convergence: Levenberg-Marquardt approximation of the Hessian with weighted elliptic regularization, backtracking line search based on the l(1) merit function, and rigorous convergence criteria. We use the proposed method to solve a range of inviscid conservation laws of varying difficulty. We show the method is able to deliver accurate solutions on coarse, high-order meshes and the SQP solver is robust and usually able to drive the first-order optimality system to tight tolerances. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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