Fuhrman, David R.2; Bingham, Harry B.2; Madsen, Per A.2; Thomsen, Per Grove5
1 Coastal, Maritime and Structural Engineering, Department of Mechanical Engineering, Technical University of Denmark2 Department of Mechanical Engineering, Technical University of Denmark3 Scientific Computing, Department of Informatics and Mathematical Modeling, Technical University of Denmark4 Department of Informatics and Mathematical Modeling, Technical University of Denmark5 Department of Applied Mathematics and Computer Science, Technical University of Denmark
This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly non-linear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neumann) techniques with matrix-based methods for formulations in both one and two horizontal dimensions. The matrix-based method is also extended to show the local de-stabilizing effects of the non-linear terms, as well as the stabilizing effects of numerical dissipation. A comparison of the relative stability of rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep-water non-linearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only moderately non-normal, suggesting that the eigenvalues are likely suitable for analysis purposes. Numerical experiments demonstrate excellent agreement with the linear analysis, and good qualitative agreement with the local non-linear analysis. The various methods of analysis combine to provide significant insight into the numerical behaviour of this rather complicated system of non-linear PDEs.
International Journal for Numerical Methods in Fluids, 2004, Vol 45, Issue 7, p. 751-773