A Conservative Fully Discrete Numerical Method for the Regularized Shallow Water Wave Equations

Abstract

The paper proposes a new, conservative fully-discrete scheme for the numerical solution of the regularised shallow water Boussinesq system of equations in the cases of periodic and reflective boundary conditions. The particular system is one of a class of equations derived recently and can be used in practical simulations to describe the propagation of weakly nonlinear and weakly dispersive long water waves, such as tsunamis. Studies of small-amplitude long waves usually require long-time simulations in order to investigate scenarios such as the overtaking collision of two solitary waves or the propagation of transoceanic tsunamis. For long-time simulations of non-dissipative waves such as solitary waves, the preservation of the total energy by the numerical method can be crucial in the quality of the approximation. The new conservative fullydiscrete method consists of a Galerkin finite element method for spatial semidiscretisation and an explicit relaxation Runge-Kutta scheme for integration in time. The Galerkin method is expressed and implemented in the framework of mixed finite element methods. The paper provides an extended experimental study of the accuracy and convergence properties of the new numerical method. The experiments reveal a new convergence pattern compared to standard Galerkin methods. E(t; η, u) := 1 2 I gη 2 + (D + η)u 2 dx, (3)

Functional‐preserving predictor‐corrector multiderivative schemes

“…The relaxation approach is not restricted to invariants and has also been extended to general functionals 𝜂 in refs. [16][17][18], resulting for example in efficient, fully-discrete, and locally entropy-stable numerical methods for computational fluid dynamics [21] and nonlinear dispersive wave equations [22][23][24].…”

A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis

“…For the time discretization we employ once again the classical four-stage Runge-Kutta method of order four where we integrate the system until time T = 10 s and with Δt = 0.05 s. A regular, unstructured mesh of the computational domain with N h = 72, 652 triangles is considered with (r, p) = (2, 3). The common for both systems solitary wave has amplitude 0.04 m, and is generated numerically using the Petviashvili method of [35] adapted appropriately in two dimensions [29]. During the experiment we record the free surface elevation at three locations (wave gauges): G 1 (−2, 0), G 2 (0, 1), G 3 (2, 0) to measure the runup around the cylinder.…”

On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations

“…where D 1 , D 2 commute [68]. Conservation of the discrete equivalent u u u T M(I −D 2 )η η η of I (η, u) can be achieved with the standard Galerkin method [48] or SBP methods using the split form…”

“…Theorem 6.1 Let D 1 be a periodic SBP first-derivative operator with diagonal mass matrix M. Then the semidiscretization (48) conserves the total mass of ε, the total mass of u, and the total energy η = E.…”

“…Fig. 8 Error growth in time for a numerically generated stegoton solution of the variable-coefficient p-system (45), using (48) with Fourier collocation in space and fifth-order Runge-Kutta in time. The conservative method uses relaxation to enforce conservation of (47) We use smooth coefficients given by…”