Introduction

To design a spectral wave model many choices and assumptions need to be made. With the development of the SWAN model, initiated by Leo Holthuijsen, Nico Booij and Roeland Ris in 1993 and published in Booij et al. (1999), a number of principles and the scope have been established. These are as follows.

• SWAN must be suitable for both nearshore and oceanic applications. This model properly accounts for deep and shallow water wave processes (e.g. wind input, white capping, quadruplets, surf breaking and triads, as discussed in Section 2.3), and employs flexible meshes (curvilinear and triangular grids) to accommodate both small- and large-scale simulations. Typically, grid sizes can vary between 20 m (to resolve small-scale features in the seabed topography) and 100 km (to resolve large-scale features in the (hurricane) wind field). The action balance equation is formulated in Cartesian coordinates, and optionally in spherical coordinates.
• The discretization of the action balance equation must be simple, robust, accurate and economical for applications in coastal waters. Therefore a finite difference approach is employed based on the so-called method of lines. This means that the choice for time integration is independent of the choice for spatial discretization. In addition, time integration is fully implicit, which implies that the employed finite difference schemes are stable (in the Von Neumann sense) for an arbitrarily large time step irrespective of grid size. These schemes need only to be accurate enough for a time step and a grid size solely determined by physical accuracy for the scale of the phenomena to be simulated.
• The finite difference schemes for propagation of wave action in both geographic and spectral spaces must comply the causality principle, which is an essential property of the hyperbolic equation. Preservation of causality ensures that wave energy propagates in the right direction and at the right speed. Causality requires that wave energy being propagated from one point to another point further downstream must pass through all the grid points on its path between them. As a consequence, propagation schemes must look for wave energy by following wave characteristics in an upwind fashion at the right speed, while satisfying a CFL criterion. In addition, a sweeping algorithm in compliance with the causality rule has been adopted.
• Moreover, the finite difference schemes must also obey the law of constant energy flux along the wave ray, which is necessary for correct wave shoaling, especially in case of rapidly varying bathymetry (e.g. seamounts, shelf breaks, main channels in estuaries, and floodplain areas in rivers).
• Finally, some measures have been employed to guarantee numerical stability at large time steps. These are the action density limiter, the frequency-dependent under-relaxation in the iterative procedure, the conservative elimination of negative energy densities (not for the QC approximation!), the refraction limiter and the Patankar rules for linearization of the nonlinear source terms.

The numerical approaches outlining the abovementioned concepts, assumptions and principles will be discussed in the following sections.