Infinite errors using different boundary conditions.

## 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.

### Keywords

- wave equations
- transparent boundary condition
- boundary element method
- non-standard boundary conditions
- finite difference method

## 1. Introduction

We consider a combination of two linearized typical dispersive partial differential equations that model solitary waves and all interactions between them, given as follows

such that

The Boundary Element Method (BEM), also known as the Boundary Integral Equation Method (BIEM), is an alternative deterministic method that incorporates a mesh located, only, at domain boundaries and therefore attractive for free surface problems. There are two types of BEM, the direct BEM which requires a closed boundary so that the physical variables (e.g. pressure and normal velocity in acoustics) can only be considered from one side of the surface (interior or exterior), while the indirect (IBEM) can consider both sides of the surface and does not need a closed surface. In the first part of this chapter, we use this technique of BEM to derive the BC to the Eq. (1) in the domain [0, * T*] × [

*]. More precisely, we are going to introduce the BEM to establish BC satisfied by the Eq. (1) on two interface points*a, b

*and*a

*by solving the same equation in the complementary domain*b

*] ×*T

Despite the meshing effort is limited and the system matrices are smaller, the BEM also has disadvantages over the Finite Element Method or Difference Finite Method. In fact, the BEM matrices are mostly populated with complex coefficients. Furthermore, singularities may arise in the solution. These deteriorate the efficiency of the solution and must be prevented [2].

The outline of this chapter is organized as follows. In section 2, we describe the BEM for the mixed BBM-KdV equation [3]. Next, we discuss the special case of the BBM equation and give the approximation of the resulting equation Finite Difference Method. Section 3 presents briefly another method to derive boundary conditions for BBM equation called NSBC introduced in [4]. Finally in section 4, comparison of both methods is given with numerical experiments to highlight the transparency of both BC obtained in sections 2 and 3.

## 2. Boundary element method for the mixed BBM-KdV equation

Being in one dimensional space,

The main object of this section is to prove the following result.

Lemma 2.1 The solution of the evolution Eq. (3) satisfies the following integral equations

where

** Proof.**We apply the Laplace transformation with respect to the time variable

where

We obtain

which is a cubic ordinary differential equation whose solutions are of the form are given explicitly by

where

The three solutions are given by

where the complex

Assume that

In fact, we consider the change of variable

Hence under the condition

Now back to Eq. (7), for

Idem for

As

Then the Eq. (13) becomes in terms of

Now applying the inverse Laplace transform to Eqs. (8) and (10), we infer

Therefore, we get the following result describing the problem in the bounded domain satisfied by the restriction on

Theorem 1.1 Let

where

We emphasize that those boundary conditions strongly depend on

In this case, we obtain convolution products with Bessel functions after the Laplace inverse transformation as follows

where we have used the expressions

and the notations

Recall that the Bessel functions can be defined by the following integrals

From this, we may compute

Thus the boundary operator

Next, we propose an approximation, always for the case

### 2.1 Numerical approximation

This subsection is devoted to the numerical approximation of the obtained IBVP (17) for

Let

#### 2.1.1 Approximation of the governing equation

We describe a discretization for the BBM equation by the Crank-Nicholson time scheme as follows

For the space finite difference scheme, we use the approximations

The fully discretization then writes,

#### 2.1.2 Approximation of the boundary condition

The constructed boundary conditions (BC) contains time convolutions that are non-local and introduces many difficulties, for example, using a direct implementation leads to long and low accuracy. Several techniques have been used to overcome these problems by trying to localize the BC, see [5, 6, 7, 8] for more details. The resulting localized BC are easy to implement and more efficient but tends to depend sensitively on the initial data. In our case, we utilize the Discrete Galerkin Method. The BC are formulated as Fredholm integral equations of second kind. The basic idea is to write the boundary condition on (20) in the form

where, the introduced Kernels

where

The Eq. (27) is a Fredholm integral equation of second kind, where

The approximation of the BC is now given by

that is

For the seek of simplicity, we rewrite the integral terms of (31) and (32) in the form

such that

The second integral is more complicated since it involves two composing integrals, using Gauss-Legendre quadrature twice yields

and the remained integral is approximated by

From approximations (33)–(35), the numerical solution on the interface of (17) can be given by

We accomplish this by simply adding (36) and (37) to the discretization of the interior governing Eq. (22). We obtain an implicit scheme that we illustrate by the following system in matrix form

with

and

where the discretization constants are

## 3. Non standard boundary conditions for the BBM equation

In [4], we have presented a new method to derive transparent boundary conditions for the BBM equation. These boundary conditions have the advantage of being local in time but needs an additional function construct the BC which means bigger system to be solved. We recall that the problem designed to be the restriction in

## 4. Numerical examples

We take an initial condition as solitary wave like function locally supported in

Let

Table 1 shows that both methods give a good approximation of the restriction to

BC | CPU time(s) | ||||
---|---|---|---|---|---|

4 | |||||

9 | |||||

15 | |||||

5 | |||||

39 | |||||

62 | |||||

80 | |||||

50 | |||||

436 | |||||

1040 | |||||

3205 | |||||

1015 |

We also plot in Figure 1, captions at different times of either reference solution and approximated solutions using NSBC and

## 5. Conclusion

We have compared two methods of deriving and approaching boundary conditions for the BBM equation. We presented the BEM for a general equation that is the mixed BBM-KdV equation and that shows the hardness to put easy implemented BC. Furthermore, being non local in time, BC seems to be low accurate and slowly convergent as presented in numerical example. However, this point opens many possibilities trying to improve the accuracy of such BC whether by improving the approximation of convolution product, that comes from Inverse Laplace transformation, via quadrature or exploring a numerical equivalent to such operation such as