ADMOS 2023

A hybrid adaptive method for initial-boundary value problems

  • Mattsson, Ken (Uppsala University)
  • Ahn Dao, Tuan (Uppsala University)
  • Eriksson, Gustav (Uppsala University)
  • Stiernström, Vidar (Uppsala University)

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It is well-known that higher-order methods (as compared to lower order accurate methods) capture transient phenomena more efficiently since they allow a considerable reduction in the degrees of freedom for a given error tolerance. In particular, high-order finite difference methods (HOFDMs) are ideally suited for problems of this type. For long-time simulations, it is imperative to use finite difference approximations that do not allow growth in time if the PDE does not allow growth---a property termed time stability. Achieving time-stable HOFDM has received considerable past attention. A robust and well-proven high-order finite difference methodology, for well-posed initial boundary value problems (IBVP), is to combine summation-by-parts (SBP) operators and either the simultaneous approximation term (SAT) method, or the projection method to impose boundary conditions. Other examples of discrete operators with the SBP property include spectral collocation, finite volume methods, and continuous Galerkin FE. The SBP-SAT and SBP-Projection methods naturally extends to multi-block geometries. Thus, problems involving complex domains or non-smooth geometries are easily amenable to the approach. These methods can also handle non-conformal interfaces, allowing adaptive grids. The SBP-SAT and SBP-Projection methods also allow for a hybrid coupling of different schemes having SBP property. A hybrid SBP-SAT method to couple HOFDM and FEM in a nonconforming multiblock fashion is presented. Our most recent results indicate that the less well-known SBP-Projection method has some important advantages as compared to the now relatively mature SBP-SAT method. One of the more obvious advantages with the SBP-Projection method is that it exactly mimics the stability properties of the underlying well-posed IBVP, without tuning of parameters. The SBP-Projection method only requires the discrete operators to have a SBP property. In the present study we will show how the SBP-Projection method can be employed to ensure time-stability and efficiency in the framework of well-posed IBVP. As a proof of concept we will show results from various wave dominated problems, including the Navier-Stokes equations, the elastic wave equation, the acoustic wave equation, and flexural-gravity waves in ice-covered oceans. Some novel results of FD-FD, as well as hybrid FE-FD coupling using the SBP-Projection method on non-conformal interfaces will also be presented.