A discontinuous Galerkin approach for conservative modeling of fully nonlinear and weakly dispersive wave transformations

Abstract

This work extends a robust second-order Runge-Kutta Discontinuous Galerkin (RKDG2) method to solve the fully nonlinear and weakly dispersive flows, within a scope to simultaneously address accuracy, conservativeness, cost-efficiency and practical needs. The mathematical model governing such flows is based on a variant form of the Green-Naghdi (GN) equations decomposed as a hyperbolic shallow water system with an elliptic source term. Practical features of relevance (i.e. conservative modelling over irregular terrain with wetting and drying and local slope limiting) have been restored from an RKDG2 solver to the Nonlinear Shallow Water (NSW) equations, alongside new considerations to integrate elliptic source terms (i.e. via a fourth-order local discretization of the topography) and to enable local capturing of breaking waves (i.e. via adding a detector for switching off the dispersive terms). Numerical results are presented, demonstrating the overall capability of the proposed approach in achieving realistic prediction of nearshore wave processes involving both nonlinearity and dispersion effects within a single model.

Performance study of the multiwavelet discontinuous Galerkin approach for solving the Green‐Naghdi equations

“…Figure 5 shows the water depth and flow discharge errors computed at t = 5 seconds with both MWDG-GN and DG-GN solvers for all the settings except the coarsest one with {1, 7}, which was not included to save space. 11 This implies that the adaptive MWDG-GN solver would require smaller threshold values compared to an adaptive MWDG-NSW solver 47,49 to accommodate finer resolution needs. As compared to a NSW solver for numerical modeling of nondispersive flows, a GN solver necessitates finer grid resolution to ensure capturing both dispersive and nonlinear features.…”

“…2 The nonlinearity parameter is another related identifier, which is defined as the ratio of the wave amplitude scale to the water depth scale, = a/h 0 . [9][10][11] DG methods are becoming increasingly popular in solving BT equations. Removing this assumption (ie, let = O(1)) while keeping all the O( ) terms gives the so-called Green-Naghdi (GN) equations.…”

“…Most of the BT equations impose a smallness amplitude assumption as = O( 2 ), which is too restrictive for many applications in nearshore areas. 7,[11][12][13][14][15][16][17][18][19] However, the runtime cost of DG methods is high, given their demands for storage and evolution of local degrees of freedom within each computational cell and their restrictive CFL condition when applied with explicit Runge-Kutta (RK) time stepping. [3][4][5] The GN equations share the same characteristics of other BT models.…”

“…An existing uniform mesh DG solver for the GN equations (DG-GN) 11 is extended to adaptive form, following the MWDG method introduced in the work of Kesserwani et al 47 that is applied to the NSW equations (MWDG-NSW). An existing uniform mesh DG solver for the GN equations (DG-GN) 11 is extended to adaptive form, following the MWDG method introduced in the work of Kesserwani et al 47 that is applied to the NSW equations (MWDG-NSW).…”

“…9-11 DG methods are becoming increasingly popular in solving BT equations. 7,[11][12][13][14][15][16][17][18][19] However, the runtime cost of DG methods is high, given their demands for storage and evolution of local degrees of freedom within each computational cell and their restrictive CFL condition when applied with explicit Runge-Kutta (RK) time stepping. These costs would even be higher when modeling wave propagation and transformation in coastal areas, where the multitude of spatial and temporal scales further increase the wave feature and complexity.Classical adaptive mesh refinement (AMR) techniques were initially used in an attempt to reduce fine resolution costs by adapting the mesh resolution.…”