Physics-Capturing Discretizations for Spectral Wind-Wave Models

Marcel Zijlema

doi:10.3390/fluids6020052

Abstract

This paper discusses the discretization methods that have been commonly employed to solve the wave action balance equation, and that have gained a renewed interest with the widespread use of unstructured grids for third-generation spectral wind-wave models. These methods are the first-order upwind finite difference and first-order vertex-centered upwind finite volume schemes for the transport of wave action in geographical space. The discussion addresses the derivation of these schemes from a different perspective. A mathematical framework for mimetic discretizations based on discrete calculus is utilized herein. A key feature of this algebraic approach is that the process of exact discretization is segregated from the process of interpolation, the latter typically involved in constitutive relations. This can help gain insight into the performance characteristics of the discretization method. On this basis, we conclude that the upwind finite difference scheme captures the wave action flux conservation exactly, which is a plus for wave shoaling. In addition, we provide a justification for the intrinsic low accuracy of the vertex-centred upwind finite volume scheme, due to the physically inaccurate but common flux constitutive relation, and we propose an improvement to overcome this drawback. Finally, by way of a comparative demonstration, a few test cases is introduced to establish the ability of the considered methods to capture the relevant physics on unstructured triangular meshes.

Highlights

The numerical solution of partial differential equations (PDEs) is traditionally sought by a discretization method, such as the finite difference, finite volume, or finite element method, aimed towards constructing a scheme that is consistent to some order of accuracy, while maintaining the numerical stability
This implies that certain topological structures that are embedded invisibly in the PDEs may not be sufficiently represented in the conventional discretization process
The discretization methods as outlined in [36] were reconstructed in an effort to point out the presence or the lack of wave action flux conservation

Summary

Introduction

The numerical solution of partial differential equations (PDEs) is traditionally sought by a discretization method, such as the finite difference, finite volume, or finite element method, aimed towards constructing a scheme that is consistent to some order of accuracy, while maintaining the numerical stability. The inherent assumptions underlying the above approach are smoothness and differentiability of the PDEs imposed by the limit process This implies that certain topological structures that are embedded invisibly in the PDEs may not be sufficiently represented in the conventional discretization process. A strict control of the discretization error, as happened with many numerical methods (e.g., high-order regularization techniques and high-resolution TVD and WENO reconstructions for hyperbolic conservation laws), can not guarantee that the essential physics of the underlying problem will capture properly. This aspect becomes relevant for problems with strong nonlinearities and discontinuities

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Results

Conclusion