2 research outputs found
An optimization-based approach for high-order accurate discretization of conservation laws with discontinuous solutions
This work introduces a novel discontinuity-tracking framework for resolving
discontinuous solutions of conservation laws with high-order numerical
discretizations that support inter-element solution discontinuities, such as
discontinuous Galerkin methods. The proposed method aims to align inter-element
boundaries with discontinuities in the solution by deforming the computational
mesh. A discontinuity-aligned mesh ensures the discontinuity is represented
through inter-element jumps while smooth basis functions interior to elements
are only used to approximate smooth regions of the solution, thereby avoiding
Gibbs' phenomena that create well-known stability issues. Therefore, very
coarse high-order discretizations accurately resolve the piecewise smooth
solution throughout the domain, provided the discontinuity is tracked. Central
to the proposed discontinuity-tracking framework is a discrete PDE-constrained
optimization formulation that simultaneously aligns the computational mesh with
discontinuities in the solution and solves the discretized conservation law on
this mesh. The optimization objective is taken as a combination of the the
deviation of the finite-dimensional solution from its element-wise average and
a mesh distortion metric to simultaneously penalize Gibbs' phenomena and
distorted meshes. We advocate a gradient-based, full space solver where the
mesh and conservation law solution converge to their optimal values
simultaneously and therefore never require the solution of the discrete
conservation law on a non-aligned mesh. The merit of the proposed method is
demonstrated on a number of one- and two-dimensional model problems including
2D supersonic flow around a bluff body. We demonstrate optimal
convergence rates in the norm for up to polynomial
order and show that accurate solutions can be obtained on extremely
coarse meshes.Comment: 40 pages, 23 figures, 1 tabl
Implicit shock tracking using an optimization-based high-order discontinuous Galerkin method
A novel framework for resolving discontinuous solutions of conservation laws,
e.g., contact lines, shock waves, and interfaces, using implicit tracking and a
high-order discontinuous Galerkin (DG) discretization was introduced in [38].
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 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 space and 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, 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.Comment: 35 pages, 20 figure