Analysis and application of the discontinuous Galerkin method to the RLW equation
© Hozman and Lamač; licensee Springer. 2013
Received: 14 December 2012
Accepted: 23 April 2013
Published: 7 May 2013
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 semi-discretization 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.
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 ) to provide an alternative description of nonlinear dispersive waves to the Korteweg-de 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 pulse-like solitary wave solutions or periodic waves; see . It governs various physical phenomena in disciplines such as nonlinear transverse waves in shallow water, ion-acoustic 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 , over collocation methods [4, 5], to finite element approaches [6, 7], or Galerkin methods , and in references cited therein.
In this paper, we present a semi-implicit 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 semi-discretization uses higher-order piecewise polynomial discontinuous approximation on arbitrary meshes; for a survey, see [9, 10] and . Among several variants of DG methods, we deal with the nonsymmetric variant interior penalty Galerkin discretizations; see . 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 . Consequently, we extend the results from , and the attention is paid to the a priori error analysis of the method with the aid of standard techniques introduced in  and .
The rest of the paper is organized as follows. The problem formulation and its variational reformulation are given in Section 2. Discretization, including space semi-discretization 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
The whole system (1)-(4) was found to have single solitary or periodic traveling wave solutions; for details, see .
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.
for a norm in by , for a seminorm in by and for a norm in by . It is a known fact that is a norm on equivalent to
where symbol stands for the function on Ω such that , and function in (7) represents the physical flux.
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 periodic-type solutions with the aid of Sobolev spaces of periodic functions with period satisfying .
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
Let () be a family of the partitions of the closure of the domain Ω into N closed mutually disjoint subintervals with length and the symbol stands for an index set . Then we call a partition with a spatial step and interval an element. By we denote the set of all partition nodes of Ω, i.e., . Further, we label by the set of all inner nodes. Obviously, .
We additionally assume that the partitions satisfy the following condition.
The condition (12) in fact allows to control a level of the mesh refinement if adapted meshes are used.
where and denote the standard norm and the seminorm on the Sobolev space , .
where denotes the space of all polynomials of degree on , .
By convention, we also extend the definition of jump and mean value for endpoints of Ω, i.e., , , and . In case that are arguments of or , we usually omit these arguments , and write simply and , respectively.
3.1 DG semi-discrete formulation
where and the choice of and for boundary points has to satisfy the prescribed Dirichlet boundary conditions; for more details, see .
In what follows, we shall assume that the numerical flux has the following properties.
One can see that the numerical flux H given by (21) satisfies conditions (H2) and (H3) and is Lipschitz-continuous 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 order to guarantee the stability of the numerical scheme. In our case, where this stabilization is added with a positive sign (+), we speak of the nonsymmetric interior penalty Galerkin method.
which replaces the inter-element discontinuities and guarantees the fulfillment of the prescribed boundary conditions.
which is bilinear due to (19) and (25). Consequently, we can here define the semi-discrete solution of problem (8).
3.2 Semi-implicit 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 , we have proposed a semi-implicit time discretization based on the backward Euler scheme with the linearized convection form which is suitable for avoiding the strong time step restriction of explicit time schemes as well as for the preservation of linear algebraic problems at each time level.
where and represent the corresponding linearized and nonlinear parts of the original numerical flux given by (21); for more details, see . One can easily observe that the form is linear in its second and third argument and the form is in fact an original convection form (20) with half the amount of the physical flux but from the previous time level.
The fully discrete solution of problem (18) via the aforementioned semi-implicit approach is defined in following way.
Discrete problem (33) is equivalent to a system of linear algebraic equations at each time instant . In what follows, we shall be concerned with the analysis of method (33).
Lemma 1 Discrete problem (33) has a unique solution.
Hence, equation (34) has a unique solution . □
4 A priori error analysis
For error analysis and in experiments, we consider for all . Thus we denote . Now we would like to analyze the error estimates of the approximate solution , , obtained by method (33). For simplicity, we consider a uniform partition , , of the time interval with time step , where is an integer.
For next estimates, we use the following lemmas.
where c is a generic constant independent of h and τ.
Proof The proof of these standard estimates can be found, for instance, in . □
where c is a generic constant independent of h and τ.
Proof Again, one can find the proof of these estimates in . □
In order to finish our estimates, we require a fulfillment of the following technical assumption.
(T1) There exists such that .
If assumption (T) is fulfilled, then . Thus we can also reformulate assumption (T) so that .
where we used a straightforward estimate . We can notice that due to the presence of the factor μ in front of the function on the left-hand side of (65), we lost μ in denominators of .
Now we are ready to formulate the main theorem.
where is defined by (60).
Remark 3 Theorem 1 implies that the error of our method is in both energy and -norm. However, as we will see in the next section, the error estimate in the -norm is suboptimal with respect to h.
Remark 4 The dependency C on μ in the expression (68) (choice of θ depends on μ) can be removed by applying the so-called continuous mathematical induction mentioned in . This is useful namely in the cases when convection terms dominate, i.e., . Consequently, in these cases assumption (T) can be weakened to a CFL-like condition for suitable .
5 Numerical experiments
In this section we shall numerically verify the theoretical a priori error estimates of the proposed semi-implicit method (33) for the cases of propagation of both a single solitary wave and periodic waves.
5.1 Single solitary case
which represents a single solitary wave of amplitude 3c, traveling with the velocity in a positive x-direction and located initially at the point . The initial condition is extracted from the exact solution (74) and homogeneous Dirichlet boundary conditions are set.
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 . Numerical experiments are carried out with the use of piecewise linear () approximations on five consecutive uniformly refined meshes having 125, 250, 500, 1,000 and 2,000 elements.
Single solitary case: Computational errors in the -norm and experimental orders of convergence for approximation on a consequence of meshes at time instances t ( )
Single solitary case: Computational errors in the energy norm and experimental orders of convergence for approximation on a consequence of meshes at time instances t ( )
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 . Finally, both estimates in Theorem 1 confirm the well-know attribute of DG schemes from the class of convection-diffusion problems, cf.  and .
5.1.2 Convergence with respect to τ
Secondly, we verify experimentally the convergence of the method in the -norm and the energy norm with respect to time step τ. In order to restrain the discretization errors with respect to h, we use a fine mesh with 2,000 elements with piecewise linear approximation.
Single solitary case: Computational errors in the -norm and the energy norm for approximation with respect to time step ( )
5.1.3 Invariant conservation quantities
assessing the accuracy of the method by measuring the difference between the numerical and analytic solutions and u, respectively.
Single solitary case: Computed invariant quantities and errors in the -norm and the -norm ( , )
ref. meth.  (h = 0.125, τ = 0.1)
ref. meth.  (h = 0.1, τ = 0.1)
ref. meth.  (h = 0.8, τ = 0.1)
ref. meth.  (h = 0.125, τ = 0.1)
ref. meth.  (h = 0.05, τ = 0.1)
analytical val. (c = 0.1, μ = 1.0)
5.2 Periodic case
where is a complete elliptic integral of the first kind, see . The limit implies that the periodic behavior reduces to the propagation of a single solitary wave.
In order to compute the periodic case on approximately the same space-time domain as in the single solitary case, we again set the parameter values , , and the parameter k is experimentally set up as to have periods and .
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 h-convergence in the periodic case is investigated on a sequence of five successive refined grids partitioning the considered problem domain . The choice of time step is again small enough to suppress the influence of time discretization errors, and the computations are performed by piecewise linear approximations, subsequently.
Periodic case: Computational errors in the -norm and experimental orders of convergence for approximation on a consequence of meshes at time instances t ( )
Periodic case: Computational errors in the energy norm and experimental orders of convergence for approximation on a consequence of meshes at time instances t ( )
5.2.2 Convergence with respect to τ
Periodic case: Computational errors in the -norm and the energy norm for approximation with respect to time step ( )
From the presented numerical results in Sections 5.1.1-5.1.2 and 5.2.1-5.2.2, 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 , all these three invariants of motion remain conserved and equal to their original values that are well-determined analytically at .
Periodic case: Computed invariant quantities and errors in the -norm and the -norm ( , )
analytical val. (k = 0.63048)
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 semi-implicit time discretization with suitable linearization of convective terms. Under some additional assumptions, we have derived a priori error estimates, namely in the -norm and in the energy norm. On the other hand, the presented numerical experiments for single solitary as well as periodic cases signal a better behavior of the experimental -order of convergence, which is expected to be asymptotically for piecewise linear approximations with a nonsymmetric variant of interior penalty Galerkin discretizations. In the case of the energy norm, we obtain the optimal experimental order of convergence.
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.
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.
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