ABSTRACT
V. A. Dougalis, Mathematics Department, University of Athens, 15784 Zographou, Greece and Foundation for Research and Technology-Hellas, Institute of Applied and Computational Mathematics, 71110 Heraklion, Greece, doug@math.uoa.gr
D. E. Mitsotakis, Mathematics Department, University of Athens, 15784 Zographou, Greece and Foundation for Research and Technology-Hellas, Institute of Applied and Computational Mathematics, 71110 Heraklion, Greece, dmitsot@gmail.com
In this chapter we shall be mainly concerned with the numerical solution of the Boussinesq systems of equations that occur in water wave theory. These systems, derived in their general form in [11], may be written in nondimensional, unscaled variables as
ηt + ux + (ηu)x + auxxx − bηxxt = 0, ut + ηx + uux + cηxxx − duxxt = 0, (3.1)
where η = η(x, t), u = u(x, t) are real functions defined for x ∈ IR and t ≥ 0. In addition
a = 12 (θ 2 − 13 )ν, b = 12 (θ2 − 13 )(1− ν),
c = 12 (1− θ2)µ, d = 12 (1− θ2)(1− µ), (3.2)
where ν and µ are constants and 0 ≤ θ ≤ 1. Our main goal in this paper is to review the existing mathematical theory for the systems (3.1) and solve them numerically for specific choices of a, b, c, d in order to study interesting features of their solutions.
