SLATE-based solvers for semi-implicit hybridised DG methods in fluid dynamics

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Our research focuses on Numerical Weather Prediction (NWP), a problem in fluid dynamics which requires the use of high performance computing. At the core of this we must solve the Navier-Stokes equations, which are a non-linear hyperbolic system of PDEs. One simplification of this system is Shallow Water Equations (SWE).

Unlike a conforming finite element method, the constraint of continuity between mesh cells is relaxed for a DG method. This is replaced with a numerical flux between cells.

For a higher order DG method, a higher degree polynomial basis is used to approximate the solution, which leads to higher accuracy.

The system of equations we solve are the shallow water equations: \begin{align} \partial_t \phi_S + c_g\nabla\cdot{U} &= 0\\ \partial_t U + c_g\nabla{\phi_S} &= -f(\hat{k}\times U) \end{align}

The first equation describes the change in potential height $\phi_S$ and the second describes the change in momentum $U$.

These equations describe the motion of a shallow liquid in a given domain.

Using the a discontinuous Galerkin method we can discretise the PDE above to obtain a block matrix equation. Using linear algebra libraries this can now be solved on a computer.

The coupling between the blocks in the DG matrix can be eliminated by using a hybridisation technique, at the cost of increasing the matrix size. However, by using these additional degrees of freedom we can still improve the solve times.

Using the Firedrake framework, which uses code generation to solve PDEs. Problems are expressed in the UFL language which is automatically turned into parallel C code and executed.

Numerical results showing the effectiveness of the solvers.

We are currently in the process of adding bathymetry and modifying the solver to work with the fully non-linear equations.

The code we are developing also allows us to solve the shallow water equations using DG discretisations in a spherical domain.

  • J. S. Hesthaven and T. Warburton.
    Nodal discontinuous Galerkin methods: algorithms, analysis, and applications.
    Springer Science & Business Media, 2007.
  • B. Cockburn.
    Discontinuous galerkin methods.
    ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 83(11):731–754, 2003.
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Jack Betteridge is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/L015684/1.