A new vertex-based limiting approach for nodal discontinuous Galerkin methods on arbitrary unstructured meshes

Development of an efficient wetting and drying treatment for shallow‐water modeling using the quadrature‐free Runge‐Kutta discontinuous Galerkin method

“…To preserve the numerical stability of the scheme, the vertex‐based slope limiter from Li and Zhang 43 is applied to the entire domain to eliminate the numerical oscillations. This slope limiter is favorable for RKDG schemes with nodal basis functions and applies to the conservative variables at each interior RK stage.…”

“…This WD method is similar to the multiwavelet‐based approach from Gerhard et al, 41 which also utilized p ‐adaptive methods to treat transition elements; however, we avoid using complex procedures associated with refining results into subcells to increase computational efficiency. Moreover, recent studies have reported that vertex‐based reconstruction methods in the slope limiters are more accurate than traditional cell‐based reconstruction methods on the unstructured grids, as the computational stencil is much larger 43,48 . Hence, we introduced the vertex‐based linear reconstruction approach in the WD treatment, and several modifications are also made to preserve the well‐balanced and positivity‐preserving properties.…”

“…where the threshold ξ w is a small positive value to provide robustness to the WD method. Then, we introduce the vertex‐based linear reconstruction approach to modify the depth and momentum in those transition elements 43 …”

“…This method employs arbitrarily high‐order polynomials in wet elements and linear solutions in the transition elements that can simultaneously provide accurate and robust results in wet and transition regions. Recently, several studies have reported that vertex‐based linear reconstruction methods are more accurate than conventional element‐based methods on unstructured meshes 42,43 . Therefore, we modify the approach from Li and Zhang 43 and introduced the vertex‐based method for transition element reconstruction, which is suitable for unstructured triangular meshes.…”

“…Recently, several studies have reported that vertex‐based linear reconstruction methods are more accurate than conventional element‐based methods on unstructured meshes 42,43 . Therefore, we modify the approach from Li and Zhang 43 and introduced the vertex‐based method for transition element reconstruction, which is suitable for unstructured triangular meshes. Based on the methods of Xing et al, 30 we proposed a restricted time step for the quadrature‐free RKDG scheme to preserve the positivity of water depth, which has a similar form but is less restricted.…”