A posteriori finite-volume local subcell correction of high-order discontinuous Galerkin schemes for the nonlinear shallow-water equations. (English) Zbl 07517733
Summary: We design and analyze a new discretization method for the nonlinear shallow water equations, which is based on an equivalent representation of arbitrary high-order Discontinuous Galerkin (DG) schemes through piecewise constant modes on a sub-grid, together with a selective a posteriori local correction of the sub-interface reconstructed flux. This new approach, based on [F. Vilar, J. Comput. Phys. 387, 245–279 (2019; Zbl 1452.65251)], allows to combine at the subcell scale the excellent robustness properties of the Finite-Volume (FV) lowest-order method and the high-order accuracy of the DG method. For any order of polynomial approximation, the resulting algorithm is shown to: (i) accurately handle strong shocks with no robustness issues; (ii) ensure the preservation of the water height positivity at the subcell level; (iii) preserve the class of motionless steady states (well-balancing); (iv) retain the highly accurate subcell resolution of DG schemes. These assets are numerically illustrated through an extensive set of test-cases, with a particular emphasize put on the use of very-high order polynomial approximations on coarse grids.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 76Mxx | Basic methods in fluid mechanics |
| 76Bxx | Incompressible inviscid fluids |
Keywords:
non-linear shallow water equations; discontinuous Galerkin scheme; local a posteriori subcell FV correction; well-balancing; positivity preservation; arbitrary high-orderCitations:
Zbl 1452.65251Software:
MOODReferences:
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