{
  "id": "2101.08913",
  "version": "v1",
  "published": "2021-01-22T01:35:55.000Z",
  "updated": "2021-01-22T01:35:55.000Z",
  "title": "Implicit shock tracking for unsteady flows by the method of lines",
  "authors": [
    "Andrew Shi",
    "Per-Olof Persson",
    "Matthew Zahr"
  ],
  "comment": "24 pages, 12 figures",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "A recently developed high-order implicit shock tracking (HOIST) framework for resolving discontinuous solutions of inviscid, steady conservation laws [41, 43] is extended to the unsteady case. Central to the framework is an optimization problem which simultaneously computes a discontinuity-aligned mesh and the corresponding high-order approximation to the flow, which provides nonlinear stabilization and a high-order approximation to the solution. This work extends the implicit shock tracking framework to the case of unsteady conservation laws using a method of lines discretization via a diagonally implicit Runge-Kutta method by \"solving a steady problem at each timestep\". We formulate and solve an optimization problem that produces a feature-aligned mesh and solution at each Runge-Kutta stage of each timestep, and advance this solution in time by standard Runge-Kutta update formulas. A Rankine-Hugoniot based prediction of the shock location together with a high-order, untangling mesh smoothing procedure provides a high-quality initial guess for the optimization problem at each time, which results in Newton-like convergence of the sequential quadratic programing (SQP) optimization solver. This method is shown to deliver highly accurate solutions on coarse, high-order discretizations without nonlinear stabilization and recover the design accuracy of the Runge-Kutta scheme. We demonstrate this framework on a series of inviscid, unsteady conservation laws in both one- and two- dimensions. We also verify that our method is able to recover the design order of accuracy of our time integrator in the presence of a strong discontinuity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-22T01:35:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unsteady flows",
      "optimization problem",
      "high-order implicit shock tracking",
      "unsteady conservation laws",
      "nonlinear stabilization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}