Abstract
In this work, our main purpose is to develop of a sufficiently robust, accurate and efficient numerical scheme for the solution of the regularized long wave (RLW) equation, an important partial differential equation with quadratic nonlinearity, describing a large number of physical phenomena. The crucial idea is based on the discretization of the RLW equation with the aid of a combination of the discontinuous Galerkin method for the space semidiscretization and the backward difference formula for the time discretization. Furthermore, a suitable linearization preserves a linear algebraic problem at each time level. We present error analysis of the proposed scheme for the case of nonsymmetric discretization of the dispersive term. The appended numerical experiments confirm theoretical results and investigate the conservative properties of the RLW equation related to mass, momentum and energy. Both procedures illustrate the potency of the scheme consequently.
PACS Codes: 02.70.Dh, 02.60 Cb, 02.60.Lj, 03.65.Pm, 02.30.f.
MSC: 65M60, 65M15, 65M12, 65L06, 35Q53.
Keywords:
discontinuous Galerkin method; regularized long wave equation; backward Euler method; linearization; semiimplicit scheme; a priori error estimates; solitary and periodic wave solutions; experimental order of convergence1 Introduction
We are concerned with a proposal of a sufficiently robust, accurate and efficient numerical method for the solution of scalar nonlinear partial differential equations. As a model problem, we consider a regularized long wave (RLW) equation firstly introduced by Peregrine (in [1]) to provide an alternative description of nonlinear dispersive waves to the Kortewegde Vries (KdV) equation. As a consequence of this, the RLW can be observed as a special class of a family of KdV equations.
The RLW equation contains a quadratic nonlinearity and exhibits pulselike solitary wave solutions or periodic waves; see [2]. It governs various physical phenomena in disciplines such as nonlinear transverse waves in shallow water, ionacoustic waves in plasma or magnetohydrodynamics waves in plasma. Since the RLW equation can be solved by analytical means in special cases, the proposed numerical methods can be easily verified. Several numerical studies of the RLW equation and its modified variant have been introduced in the literature, from finite difference methods [3], over collocation methods [4,5], to finite element approaches [6,7], or Galerkin methods [8], and in references cited therein.
In this paper, we present a semiimplicit scheme for the numerical solution of the RLW equation based on an alternative approach to the commonly used methods. The discontinuous Galerkin (DG) methods have become a very popular numerical technique for the solution of nonlinear problems. DG space semidiscretization uses higherorder piecewise polynomial discontinuous approximation on arbitrary meshes; for a survey, see [9,10] and [11]. Among several variants of DG methods, we deal with the nonsymmetric variant interior penalty Galerkin discretizations; see [12]. The discretization in time coordinate is performed with the aid of linearization and the backward Euler method, sidetracking the time step restriction well known from the explicit schemes, proposed in [13]. Consequently, we extend the results from [13], and the attention is paid to the a priori error analysis of the method with the aid of standard techniques introduced in [14] and [15].
The rest of the paper is organized as follows. The problem formulation and its variational reformulation are given in Section 2. Discretization, including space semidiscretization and fully time space discretization, is considered in Section 3. The Section 4 is devoted to a priori error analysis. Finally, in Section 5, the theoretical results are illustrated by numerical tests on a propagation of a single solitary wave and experimental orders of convergence are computed for piecewise linear approximations together with invariant quantities of the RLW equation.
2 Regularized long wave equation
Let
where constant parameters
The whole system (1)(4) was found to have single solitary or periodic traveling wave solutions; for details, see [2].
Remark 1 In the case of a single solitary wave propagation, the homogeneous Dirichlet boundary conditions in (2) arise from the asymptotic behavior of the exact solution u, and the endpoints a and b are chosen large enough so that the boundaries do not affect the single solitary wave during its propagation up to final time T.
In what follows we use the standard notation for function spaces and their norms
for a norm in
A sufficiently regular solution satisfying (1)(3) pointwise is called a classical solution. Now, we are ready to introduce the concept of weak formulation. Firstly, we recall the definition of a bilinear dispersion form
where symbol
Definition 1 We say that u is a weak solution of problem (1)(3) if
Remark 2 In order to unify the definition of the weak solution (8), we consider nonhomogeneous
Dirichlet boundary conditions instead of the second parallel analysis of periodictype
solutions with the aid of Sobolev spaces of periodic functions
Further, to carry out the error analysis later, we need to specify additional assumptions on the regularity of a solution of continuous problem (1)(3). Therefore, we assume that the weak solution u is sufficiently regular, namely
Assumptions (R)
3 Discretization
Let
We additionally assume that the partitions satisfy the following condition.
Assumption (M)
The condition (12) in fact allows to control a level of the mesh refinement if adapted meshes are used.
DG methods can handle different polynomial degrees over elements. Therefore, we assign
a local Sobolev index
with the norm
and the seminorm
where
The approximate solution of variational problem (8) is sought in a finite dimensional space of discontinuous piecewise polynomial functions associated with the vector p by
where
Let us denote
By convention, we also extend the definition of jump and mean value for endpoints
of Ω, i.e.,
3.1 DG semidiscrete formulation
Now, we recall the space semidiscrete DG scheme presented in [11]. First, we multiply (1) by a test function
where forms
The crucial item of the DG formulation of the model problem is the treatment of the
convection part. We proceed analogously as in [13], where the convection terms are approximated with the aid of the following numerical
flux
where
In what follows, we shall assume that the numerical flux
Assumptions (H)
(H1)
(H2)
(H3)
One can see that the numerical flux H given by (21) satisfies conditions (H2) and (H3) and is Lipschitzcontinuous on any
bounded subset of
A particular attention should be also paid to the treatment of the dispersion terms,
which include an artificially added stabilization in the form
In the end, the semidiscrete DG scheme is completed with the weighted penalty
which replaces the interelement discontinuities and guarantees the fulfillment of the prescribed boundary conditions.
The penalty parameter function
In order to simplify the notation, we introduce the form
which is bilinear due to (19) and (25). Consequently, we can here define the semidiscrete
solution
Definition 2 We say that
3.2 Semiimplicit linearized DG scheme
In order to obtain the discrete solution, it is necessary to equip the scheme (28)
with suitable solvers for the time integration. In [13], we have proposed a semiimplicit time discretization based on the backward Euler
scheme with the linearized convection form
We now partition
which implies the splitting of a convection form in the following way:
with
where
The fully discrete solution of problem (18) via the aforementioned semiimplicit approach is defined in following way.
Definition 3 Let
Discrete problem (33) is equivalent to a system of linear algebraic equations at each
time instant
Lemma 1Discrete problem (33) has a unique solution.
Proof We rewrite problem (33) in the following way. For
where
Using the definitions (27) and (31), one can see that
Hence, equation (34) has a unique solution
4 A priori error analysis
For error analysis and in experiments, we consider
Let
for a generic constant
Then the error
Setting
On the other hand, from (18) it follows
Multiplying (43) by τ, subtracting from (42) and using again the linearity of the form
Since
For the term on the righthand side of equation (45), we use decomposition
For next estimates, we use the following lemmas.
Lemma 2Under assumptions (R) for
wherecis a generic constant independent ofhandτ.
Proof The proof of these standard estimates can be found, for instance, in [15]. □
Lemma 3Under assumptions (R), (H) and for
wherecis a generic constant independent ofhandτ.
Proof Again, one can find the proof of these estimates in [15]. □
Since
Multiplying by 2, applying the Young inequality and using the definition of the form
If we take into account that
where we denoted
Let us now introduce the socalled energy norm
and the norm
Denoting
In order to finish our estimates, we require a fulfillment of the following technical assumption.
Assumption (T)
(T1) There exists
If assumption (T) is fulfilled, then
Thus, let us assume that assumption (T) holds, then
with
and since
where we used a straightforward estimate
Now we are ready to formulate the main theorem.
Theorem 1Let assumptions (M), (H), (R) and (T) be satisfied, then there exists a constant
where
Proof Since
□
Remark 3 Theorem 1 implies that the error of our method is
Remark 4 The dependency C on μ in the expression (68) (choice of θ depends on μ) can be removed by applying the socalled continuous mathematical induction mentioned
in [19]. This is useful namely in the cases when convection terms dominate, i.e.,
5 Numerical experiments
In this section we shall numerically verify the theoretical a priori error estimates of the proposed semiimplicit method (33) for the cases of propagation of both a single solitary wave and periodic waves.
We verify numerically the convergence of the method in the
where
where
The constants
5.1 Single solitary case
The RLW equation has the following analytical single solitary wave solution given by
which represents a single solitary wave of amplitude 3c, traveling with the velocity
In order to compare our semiimplicit approach to the schemes given in [6,20] and [5], we set the parameter values
Figure 1. The 3D plot of approximation solutions of a single solitary wave (left) and corresponding isolines in spacetime domain (right).
5.1.1 Convergence with respect to h
First, we investigate the convergence of the method with respect to h. In order to restrain the discretization errors with respect to time step τ, we use a sufficiently small time step
Tables 1 and 2 show computational errors in the
Table 1. Single solitary case: Computational errors in the
Table 2. Single solitary case: Computational errors in the energy norm and experimental orders
of convergence for
Further, the results for EOC in the energy norm are in a quite good agreement with
derived theoretical estimates; in other words, this technique produces an optimal
order of convergence
5.1.2 Convergence with respect to τ
Secondly, we verify experimentally the convergence of the method in the
The computations were carried out with five different time steps τ, see Table 3. The computational error is evaluated at final time
Table 3. Single solitary case: Computational errors in the
5.1.3 Invariant conservation quantities
Similarly as in [13], we shall monitor the three conservation quantities for the propagation of the single solitary wave corresponding to mass
momentum
and energy
with respect to the run of the proposed algorithm. The analytical values for the invariants on the entire real domain are given (in [20]) by
Moreover, for the purpose of a more accurate comparison with reference results, we
introduce the discrete
assessing the accuracy of the method by measuring the difference between the numerical
and analytic solutions
Table 4 records the invariant quantities together with errors
Table 4. Single solitary case: Computed invariant quantities and errors in the
5.2 Periodic case
The family of periodic solutions of the RLW equation may be analytically written as (cf.[20])
and the parameters
where
where
In order to compute the periodic case on approximately the same spacetime domain
as in the single solitary case, we again set the parameter values
The run of the algorithm is carried out up to one time period
Figure 2. The 3D plot of approximation solutions of periodic waves (left) and corresponding isolines in spacetime domain (right).
In what follows, we shall proceed similarly as in Section 5.1 to verify the convergence and preservation of studied invariant quantities.
5.2.1 Convergence with respect to h
The hconvergence in the periodic case is investigated on a sequence of five successive
refined grids partitioning the considered problem domain
The obtained results recorded in Tables 5 and 6 illustrate the same behavior of computational errors in the
Table 5. Periodic case: Computational errors in the
Table 6. Periodic case: Computational errors in the energy norm and experimental orders of
convergence for
5.2.2 Convergence with respect to τ
The τconvergence is experimentally verified by the computations on the finest spatial
grid having 1,760 elements with piecewise linear approximation. The computations are
performed by five different time steps τ and monitored at final time of one period
Table 7. Periodic case: Computational errors in the
From the presented numerical results in Sections 5.1.15.1.2 and 5.2.15.2.2, we see that the quality of approximate solutions obtained for a single solitary case and a periodic case is quite comparable.
5.2.3 Invariant conservation quantities
Similarly as in Section 5.2.3, we monitor the preservation of invariants of mass,
momentum and energy defined by (75), (76) and (77), respectively. During the whole
period of time, in the course of which the waves propagate inside the periodic domain
The lack of similar problems in the literature caused that our experiments with periodic
waves could not be compared with other methods, thus Table 8 captures only the development of errors in the
Table 8. Periodic case: Computed invariant quantities and errors in the
6 Conclusion
We have presented and theoretically analyzed an efficient numerical method for the
solution of the RLW equation, which is based on the space dicretization by the discontinuous
Galerkin method and a semiimplicit time discretization with suitable linearization
of convective terms. Under some additional assumptions, we have derived a priori error estimates, namely
The obtained results confirm that the proposed scheme is a powerful and reliable method for the numerical solution of a nonstationary nonlinear partial differential equation such as the RLW equation.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors contributed equally, read and approved the final version of the manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors would like to express their sincere gratitude to the referees for valuable comments and helpful suggestions. JH also would like to thank P. Červenková for her assistance with elaboration of numerical experiments. This work was partly supported by the ESF Project No. CZ.1.07/2.3.00/09.0155 ‘Constitution and improvement of a team for demanding technical computations on parallel computers at TU Liberec’ and by SGS Project ‘Modern numerical methods’ financed by TU Liberec.
References

Peregrine, DH: Calculations of the development of an undular bore. J. Fluid Mech.. 25, 321–330 (1966). Publisher Full Text

Pava, JA: Nonlinear Dispersive Equations: Existence and Stability of Solitary and Periodic Travelling Wave Solutions, Am. Math. Soc., Providence (2009)

Esen, A, Kutluay, S: A finite difference solution of the regularized longwave equation. Math. Probl. Eng.. 2006, (2006) Article ID 85743

Haq, F, Islam, S, Tirmizi, IA: A numerical technique for solution of the MRLW equation using quartic Bsplines. Appl. Math. Model.. 34(12), 4151–4160 (2010). Publisher Full Text

Islam, S, Haq, S, Ali, A: A meshfree method for the numerical solution of the RLW equation. J. Comput. Appl. Math.. 223(2), 997–1012 (2009). Publisher Full Text

Chen, Y, Mei, L: Explicit multistep method for the numerical solution of RLW equation. Appl. Math. Comput.. 218, 9547–9554 (2012). Publisher Full Text

Dag, I, Ozer, MN: Approximation of the RLW equation by the least square cubic Bspline finite element method. Appl. Math. Model.. 25, 221–231 (2001). Publisher Full Text

Esen, A, Kutluay, S: Application of a lumped Galerkin method to the regularized long wave equation. Appl. Math. Comput.. 174(2), 833–845 (2006). Publisher Full Text

Cockburn, B: Discontinuous Galerkin methods for convection dominated problems. In: Barth TJ, Deconinck H (eds.) HighOrder Methods for Computational Physics, pp. 69–224. Springer, Berlin (1999)

Cockburn B, Karniadakis GE, Shu CW (eds.): Discontinuous Galerkin Methods, Springer, Berlin (2000)

Rivière, B: Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. Frontiers in Applied Mathematics, SIAM, Philadelphia (2008)

Arnold, DN, Brezzi, F, Cockburn, B, Marini, LD: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal.. 39(5), 1749–1779 (2002). Publisher Full Text

Hozman, J: Discontinuous Galerkin method for numerical solution of the regularized long wave equation. AIP Conf. Proc.. 1497, 118–125 (2012)

Dolejší, V, Feistauer, M, Hozman, J: Analysis of semiimplicit DGFEM for nonlinear convectiondiffusion problems. Comput. Methods Appl. Mech. Eng.. 196, 2813–2827 (2007). Publisher Full Text

Dolejší, V, Feistauer, M, Sobotíková, V: Analysis of the discontinuous Galerkin method for nonlinear convectiondiffusion problems. Comput. Methods Appl. Mech. Eng.. 194, 2709–2733 (2005). Publisher Full Text

Feistauer, M, Felcman, J, Straškraba, I: Mathematical and Computational Methods for Compressible Flow, Oxford University Press, Oxford (2003)

Dolejší, V, Hozman, J: A priori error estimates for DGFEM applied to nonstationary nonlinear convectiondiffusion equation. In: Kreiss G (ed.) Numerical Mathematics and Advanced Applications, pp. 459–468. Springer, Berlin ENUMATH, 2009. (2010)

Ciarlet, PG: The Finite Elements Method for Elliptic Problems, NorthHolland, Amsterdam (1979)

Kučera, V: On εuniform error estimates for singularly perturbed problems in the DG method. Numerical Mathematics and Advanced Applications 2011. 368–378 (2013)

Djidjeli, K, Price, WG, Twizell, EH, Cao, Q: A linearized implicit pseudospectral method for some model equations: the regularized long wave equations. Commun. Numer. Methods Eng.. 19, 847–863 (2003). Publisher Full Text

Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions, Dover, New York (1965)