Third-order Finite Volume/Finite Element Solution of the Fully Nonlinear Weakly Dispersive Serre Equations
This work provides a practical numerical method for solving the Serre equations, which are important for modeling weakly dispersive water waves, but the approach is incremental as it combines existing techniques.
The authors reformulated the Serre equations to eliminate mixed derivative terms and developed a third-order hybrid finite volume/finite element scheme. The method was validated against analytical solutions, laboratory data, and dam-break simulations, demonstrating accuracy, simplicity, and stability for discontinuous flows.
The nonlinear weakly dispersive Serre equations contain higher-order dispersive terms. This includes a mixed derivative flux term which is difficult to handle numerically. The mix spatial and temporal derivative dispersive term is replaced by a combination of temporal and spatial terms. The Serre equations are re-written so that the system of equations contain homogeneous derivative terms only. The reformulated Serre equations involve the water depth and a new quantity as the conserved variables which are evolved using the finite volume method. The remaining primitive variable, the velocity is obtained by solving a second-order elliptic equation using the finite element method. To avoid the introduction of numerical dispersion that may dominate the physical dispersion, the hybrid scheme has third-order accuracy. Using analytical solutions, laboratory flume data and by simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including discontinuous flows.