Dispersive wave runup on non-uniform shores

The simulation of water waves in realistic and complex environments is a very challenging problem. Most of the applications arise from the areas of coastal and naval engineering, but also from natural hazards assessment. These applications may require the computation of the wave generation [5,12], propagation [17], interaction with solid bodies, the computation of long wave runup [16,18] and even the extraction of the wave energy [15]. Issues like wave breaking, robustness of the nu-merical algorithm in wet-dry processes along with the validity of the mathematical models in the near-shore zone are some basic problems in this direction [11]. During past several decades the classical Nonlinear Shallow Water Equations (NSWE) have been essentially employed to face these problems [7]. Mathematically, these equations represent a system of conservation laws describing the propagation of infinitely long waves with a hydrostatic pressure assumption. The wave breaking phenomenon is commonly assimilated to the formation of shock waves (or hydraulic jumps) which is a common feature of hyperbolic PDEs. Consequently, the finite volume (FV) method has become the method of choice for these problems due to its excellent intrinsic conservative and shock-capturing properties [3,7].

In the present article we report on recent results concerning the extension of the finite volume method to dispersive wave equations steming essentially from water wave modeling [14,4,6].

Consider a cartesian coordinate system in two space dimensions (x, z) to simplify notations. The z-axis is taken vertically upwards and the x-axis is horizontal and coincides traditionally with the still water level. The fluid domain is bounded below by the bottom z = -h(x) and above by the free surface z = η(x,t). Below we will also need the total water depth H(x,t) := h(x) + η(x,t). The flow is supposed to be incompressible and the fluid is inviscid. An additional assumption of the flow irrotationality is made as well.

In the pioneering work of D.H. Peregrine (1967) [14] the following system of Boussinesq type equations has been derived:

where u(x,t) is the depth averaged fluid velocity, g is the gravity acceleration and underscripts (u x , η t ) denote partial derivatives.

In our recent study [6] we proposed an improved version of this system which contains higher order nonlinear terms which should be neglected from asymptotic point of view and can be written in conservative variables (H, Q) = (H, Hu) as:

Obviously the linear characteristics of both systems (1), ( 2) and (3), (4) coincide since they differ only by nonlinear terms.

However, this modification has several important implications onto structural properties of the obtained system. First of all, the magnitude of the dispersive terms tends to zero when we approach the shoreline H → 0. This property corresponds to our physical representation of the wave shoaling and runup process. On the other hand, the resulting system becomes invariant under vertical translations (subgroup G 5 in Theorem 4.2, T. Benjamin & P. Olver (1982) [2]):

where d is some constant. This property is straightforward to check since we use only the total water depth variable H = h + η which remains invariant under transformation (5).

with reflective boundary conditions. In this case one needs to impose boundary conditions only in one of the two dependent variables, cf. [8]. In the case of reflective boundary conditions it is sufficient to take u(b

) where

(Here, we consider only uniform grids with

The governing equations (3), (4) can be recast in the following vector form:

where

We denote by H i and U i the corresponding cell averages. To discretize the dispersive terms in (6) we consider the following approximations:

We note that we approximate the reflective boundary conditions by taking the cell averages of u on the first and the last cell to be u 0 = u N+1 = 0. We do not impose explicitly boundary conditions on H. The reconstructed values on the first and the last cell are computed using neighboring ghost cells and taking odd and even extrapolation for u and H respectively. These specific boundary conditions appeared to reflect incident waves on the boundaries while conserving the mass.

This discretization leads to a linear system with tridiagonal matrix denoted by L that can be inverted efficiently by a variation of Gauss elimination for tridiagonal systems with computational complexity O(n), n-being the dimension of the system. We note that on the dry cells the matrix becomes diagonal since H i is zero on dry cells. For the time integration the explicit third-order TVD-RK method is used. In the numerical experiments we observed that the fully discrete scheme is stable and preserves the positivity of H during the runup under a mild restriction on the time step ∆t.

Therefore, the semidiscrete problem of ( 6) -( 7) is written as a system of ODEs in the form:

where L i is the i-th row of matrix L and F i+ 1 2 can

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