Accuracy aspects of conventional discretization methods for scalar transport with nonzero divergence velocity field arising from the energy balance equation

Abstract

We are concerned with the numerical solution of a linear transport problem with nonzero divergence velocity field that originates from the spectral energy balance equation describing the evolution of wind waves and swells in coastal seas. The discretization error of the commonly used first‐order upwind finite difference and first‐order vertex‐centered upwind finite volume schemes in one space dimension is analyzed. Smoothness of nondivergent velocity field plays a crucial role in this. No such analysis has been attempted to date for such problems. The two schemes studied differ in the manner in which they treat the scalar flux numerically. The finite difference variant is shock captured, whereas the vertex‐centered finite volume approximation employs an arithmetic mean of the velocity and appears not to be flux conservative. The methods are subsequently extended to two dimensions on triangular meshes. Numerical experiments are provided to verify the convergence analysis. The main finding is that the finite difference scheme displays optimal rates of convergence and offers higher accuracy over the finite volume scheme, regardless the regularity of the velocity field. The latter scheme notably yields convergence rates of 0.5 and 0 in L2‐norm and L∞‐norm, respectively, when the velocity field is not smooth. A test case illustrating wave shoaling and refraction over submerged shoals is also presented and demonstrates the practical importance of flux conservation.

Physics-Capturing Discretizations for Spectral Wind-Wave Models

“…Without the ambient current, it equals the wave group velocity. The importance of the effect of irregularity of this nondivergent field is highlighted in [36].…”

“…The resulting discrete equation is consistent with transport Equation (5). This scheme for unstructured meshes is the same as the first order vertex-centered upwind finite volume scheme described in [36]. Finally, the first order implicit Euler method is adopted for time discretization, since the action balance equation is known to be rather stiff [39].…”

“…The associated discretizations have been derived in the traditional manner, namely a first order upwind finite difference method for rectilinear and curvilinear grids, see [30][31][32], and for unstructured triangular meshes as proposed in [33]. Additionally, a first order vertex-centred upwind finite volume discretization for rectilinear grids is presented in [34,35] while the extension to unstructured meshes is described in [36]. On top of that, the issue of numerical accuracy of both discretization methods has been studied in great detail in [36].…”

“…Additionally, a first order vertex-centred upwind finite volume discretization for rectilinear grids is presented in [34,35] while the extension to unstructured meshes is described in [36]. On top of that, the issue of numerical accuracy of both discretization methods has been studied in great detail in [36].…”

“…The main goal of the current work is to address the derivation of the first order schemes for unstructured grids outlined in [33,36] from a different perspective by applying discrete calculus. We redevelop these schemes in a transparent way by separating the approximation of the constitutive relation for the wave action flux from exact discretization.…”