A high-order discontinuous Galerkin method with unstructured space–time meshes for two-dimensional compressible flows on domains with large deformations

Abstract

a b s t r a c tWe present a high-order accurate space-time discontinuous Galerkin method for solving twodimensional compressible flow problems on fully unstructured space-time meshes. The discretization is based on a nodal formulation, with appropriate numerical fluxes for the first and the second-order terms, respectively. The scheme is implicit, and we solve the resulting non-linear systems using a parallel Newton-Krylov solver. The meshes are produced by a mesh moving technique with element connectivity updates, and the corresponding space-time elements are produced directly based on these local operations. To obtain globally conforming tetrahedral meshes, we first derive the required conditions on a prism boundary mesh to allow for a valid local triangulation. Next, we present an efficient algorithm for finding a global mesh that satisfies these conditions. We also show how to add and remove mesh nodes, again using local constructs for the space-time mesh. Our method is demonstrated on a number of test problems, showing the high-order accuracy for model problems, and the ability to solve flow problems on domains with complex large deformations.

Simplex space‐time meshes in compressible flow simulations

“…10 The full space-time cylinder is first subdivided into space-time slabs, which are in turn discretized with prism-type elements, followed by a Delaunay triangulation of each prism-type element. A high-order discontinuous Galerkin method for compressible flows on domains with large deformations 13 was extended to three dimensions and applied to solve the flow around an ellipsoid rotating in a sphere. A third four-dimensional meshing strategy based on combinatorics was described in the thesis of Wang.…”

Weight‐adaptive isogeometric analysis for solving elastodynamic problems based on space‐time discretization approach

“…On the other hand, the space‐time discontinuous discretization approach is a powerful method, which leads to A ‐stable, higher‐order accurate solutions, especially, in the case of problems with discontinuities (or sharp gradient) in the solution . After the first applications of this method in the early 1970s, many researchers have used it to solve various time‐dependent problems …”

Parallel-In-Space-Time, Adaptive Finite Element Framework for Nonlinear Parabolic Equations

“…We extend the concept of error indicator defined at one time step to the notion of an error indicator defined over a block of time steps. There are several choices of error indicators with two of them being the average value of the error indicator across the block of time for an element e, (34) η e,avg = The former approach is advantageous for slow variations in the block and avoids frequent migrating of elements between processors, which can negatively affect scalability. The latter approach is advantageous when there are rapid changes across a few time steps.…”

“…We choose a time step of δt = 0.02 and consider N = 50. We remind the reader that each iteration consists of solving the space-time problem, constructing the time-averaged elemental refinement indicators using (34), and then refining the Downloaded 05/08/18 to 128.243.39.0. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php 3D mesh according to the indicators.…”

“…Figure 6 shows part of the refined spatial mesh after 20 refinement iterations. We remind the reader that each iteration consists of solving the space-time problem, constructing the time-averaged elemental refinement indicators using (34), and then refining the 3D mesh according to the indicators. Given that this is a moving source problem, this refinement is clearly around the location where the peak of the hill passes (centered 0.2 away from the midpoint with a thickness of 0.1).…”