Abstract
In this chapter, we are interested in the numerical resolution of the mixed BBM-KdV equation defined in unbounded domain. Boundary Element Method (BEM) are introduced to truncate the equation into a considered bounded domain. BEM uses domain decomposition techniques to construct Boundary Condition (BC) as transmission between the bounded domain and its complementary. We then present a suitable approximation of these BC using Discrete Galerkin Method. Numerical simulations are made to show the efficiency of these BC. We also compare with another method that truncates the equation from unbounded to bounded domain, called Non Standard Boundary Conditions (NSBC) which introduces new variables to catch information at the boundary and compose a system to connect all these variables in the bounded domain. Further discussions are made to highlight the advantages of each method as well as the difficulties encountered in the numerical resolution.
Highlights
We consider a combination of two linearized typical dispersive partial differential equations that model solitary waves and all interactions between them, given as follows 8 >∂tuðt, xÞ þ α∂3xxxuðt, β∂3txxuðt, þ γ∂xuðt, 1⁄4 > < uð0, xÞ 1⁄4> > : lim uðt, xÞ 1⁄4 ∣x∣!Æ∞0 u0ðxÞ 0 ðt, xÞ ∈ þ∗  x∈ t ∈ þ∗ (1)
We denote GLQi for approached solution with Boundary Element Method (BEM) and Gauss Legendre Quadrature in (2) for i ∈ f0, 1, 2g, while Non Standard Boundary Conditions (NSBC) refers to numerical solution with non standard boundary conditions given in (3)
GLQi is more expensive in CPU time when i increases than NSBC due to the presence of non local convolutions in time in the boundary condition
Summary
We consider a combination of two linearized typical dispersive partial differential equations that model solitary waves and all interactions between them, given as follows. Recent Developments in the Solution of Nonlinear Differential Equations suitable boundary conditions with no spurious reflections For this regard, we use two different techniques that are BEM and NSBC. The BEM has significant advantages over the finite element or difference methods (FEM or FDM), as there is no need for discretizing the domain n1⁄2a, b into elements It only uses infinite boundary condition and transmission condition to compute the solution at a and b as integral equations.
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