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Boundary Value Problems

, 2013:116 | Cite as

Analysis and application of the discontinuous Galerkin method to the RLW equation

  • Jiří HozmanEmail author
  • Jan Lamač
Open Access
Research
Part of the following topical collections:
  1. Proceedings of the International Congress in Honour of Professor Hari M. Srivastava

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 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.

Keywords

discontinuous Galerkin method regularized long wave equation backward Euler method linearization semi-implicit scheme a priori error estimates solitary and periodic wave solutions experimental order of convergence 

1 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 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 [2]. 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 [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 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 [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 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

Let Ω = ( a , b ) R Open image in new window be a bounded open interval and T > 0 Open image in new window be a final time. We set Q T = Ω × ( 0 , T ) Open image in new window and consider the following initial boundary value problem: Find u ( x , t ) : Q T = Ω × ( 0 , T ) R Open image in new window such that, for all T > 0 Open image in new window,
u t + u x + ε u u x μ t ( 2 u x 2 ) = 0 in  Q T , Open image in new window
(1)
u ( a , t ) = u D a ( t ) and u ( b , t ) = u D b ( t ) for all  t ( 0 , T ) , Open image in new window
(2)
u ( x , 0 ) = u 0 ( x ) for all  x Ω , Open image in new window
(3)
where constant parameters ε > 0 Open image in new window and μ > 0 Open image in new window are related to the amplitude of the wave and long-wavelength, respectively. From the mathematical point of view, problem (1)-(3) represents the regularized long wave equation equipped with a set of two generally nonhomogeneous Dirichlet boundary conditions (2) and with the initial condition u 0 : Ω R Open image in new window. These given data have to satisfy the compatibility conditions prescribed at both endpoints of the domain Ω, i.e.,
u D a ( 0 ) = u 0 ( a ) and u D b ( 0 ) = u 0 ( b ) . Open image in new window
(4)

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 Open image in new window and seminorms | | Open image in new window. Let k 0 Open image in new window be an integer and p [ 1 , ] Open image in new window. We use the well-known Lebesgue and Sobolev spaces L p ( Ω ) Open image in new window, H k ( Ω ) Open image in new window, Bochner spaces L p ( 0 , T ; X ) Open image in new window of functions defined in ( 0 , T ) Open image in new window with values in a Banach space X and the spaces C k ( [ 0 , T ] ; X ) Open image in new window of k-times continuously differentiable mappings of the interval [0, T] with values in X. By H 0 1 ( Ω ) Open image in new window we denote the subspace of all functions v H 1 ( Ω ) Open image in new window satisfying v ( a ) = v ( b ) = 0 Open image in new window. To this end, we use the following notation for a scalar product in L 2 ( Ω ) Open image in new window by
( u , v ) = Ω u v d x , u , v L 2 ( Ω ) Open image in new window
(5)

for a norm in L 2 ( Ω ) Open image in new window by 2 = ( , ) Open image in new window, for a seminorm in H 1 ( Ω ) Open image in new window by | | 1 , 2 = ( , ) Open image in new window and for a norm in H 1 ( Ω ) Open image in new window by 1 , 2 = ( , ) + ( , ) Open image in new window. It is a known fact that | | 1 , 2 Open image in new window is a norm on H 0 1 ( Ω ) Open image in new window equivalent to 1 , 2 Open image in new window

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 a ( , ) Open image in new window and a nonlinear convection form b ε ( , ) Open image in new window from [13], i.e.,
a ( u ( t ) , v ) = Ω u ( t ) x v d x , Open image in new window
(6)
b ε ( u ( t ) , v ) = Ω f ( u ( t ) ) x v d x with  f ( u ) = u + ε 2 u 2 , Open image in new window
(7)

where symbol u ( t ) Open image in new window stands for the function on Ω such that u ( t ) ( x ) Open image in new window, x Ω Open image in new window and function f ( u ) Open image in new window in (7) represents the physical flux.

Definition 1 We say that u is a weak solution of problem (1)-(3) if u L 2 ( 0 , T ; H 1 ( Ω ) ) L ( Q T ) Open image in new window such that u t L 2 ( 0 , T ; H 1 ( Ω ) ) Open image in new window and the following conditions are satisfied:
(a) u u L 2 ( 0 , T ; H 0 1 ( Ω ) ) , where  u ( t ) H 1 ( Ω )  such that (a) u ( t ) | x = a = u D a ( t )  and  u ( t ) | x = b = u D b ( t ) for a.e.  t ( 0 , T ) , (b) d d t ( u ( t ) , v ) + b ε ( u ( t ) , v ) + μ d d t a ( u ( t ) , v ) = 0 (b) v H 0 1 ( Ω )  and a.e.  t ( 0 , T ) , (c) ( u ( 0 ) , v ) = ( u 0 , v ) v H 0 1 ( Ω ) , u 0 L 2 ( Ω ) . Open image in new window
(8)

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 H λ k ( ( a , b ) ) Open image in new window with period λ > 0 Open image in new window satisfying mod ( b a , λ ) = 0 Open image in new window.

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 T h Open image in new window ( h > 0 Open image in new window) be a family of the partitions of the closure Ω ¯ = [ a , b ] Open image in new window of the domain Ω into N closed mutually disjoint subintervals I k = [ x k 1 , x k ] Open image in new window with length h k : = x k x k 1 Open image in new window and the symbol J Open image in new window stands for an index set { 1 , , N } Open image in new window. Then we call T h = { I k , k J } Open image in new window a partition with a spatial step h : = max k J ( h k ) Open image in new window and interval I k Open image in new window an element. By E h Open image in new window we denote the set of all partition nodes of Ω, i.e., E h = { x 0 = a , x 1 , , x N 1 , x N = b } Open image in new window. Further, we label by E h I Open image in new window the set of all inner nodes. Obviously, E h = E h I { a , b } Open image in new window.

We additionally assume that the partitions satisfy the following condition.

Assumption (M) T h Open image in new window are locally quasi-uniform:
C q 1 : h k C q h k I k , I k T h sharing a node . Open image in new window
(12)

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 s k N Open image in new window and a local polynomial degree p k N Open image in new window to each I k T h Open image in new window. Then we set the vectors s { s K , K T h } Open image in new window and p { p K , K T h } Open image in new window. Over the triangulation T h Open image in new window, we define the so-called broken Sobolev space corresponding to the vector s as
H s ( Ω , T h ) : = { v ; v | I k H s k ( I k ) I k T h } Open image in new window
(13)
with the norm
v H s ( Ω , T h ) : = ( I k T h v H s k ( I k ) 2 ) 1 / 2 Open image in new window
(14)
and the seminorm
| v | H s ( Ω , T h ) : = ( I k T h | v | H s k ( I k ) 2 ) 1 / 2 , Open image in new window
(15)

where H s k ( I k ) Open image in new window and | | H s k ( I k ) Open image in new window denote the standard norm and the seminorm on the Sobolev space H s k ( I k ) Open image in new window, I k T h Open image in new window.

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
S h p S h p ( Ω , T h ) : = { v ; v | I k P p k ( I k ) I k T h } , Open image in new window
(16)

where P p k ( I k ) Open image in new window denotes the space of all polynomials of degree p k Open image in new window on I k Open image in new window, I k T h Open image in new window.

Let us denote v ( x k + ) = lim ε 0 + v ( x k + ε ) Open image in new window and v ( x k ) = lim ε 0 + v ( x k ε ) Open image in new window. Then we can define the jump and average of v at inner points x k E h I Open image in new window of the domain Ω by
[ v ( x k ) ] = v ( x k ) v ( x k + ) , v ( x k ) = 1 2 ( v ( x k ) + v ( x k + ) ) . Open image in new window
(17)

By convention, we also extend the definition of jump and mean value for endpoints of Ω, i.e., [ v ( x 0 ) ] = v ( x 0 + ) Open image in new window, v ( x 0 ) = v ( x 0 + ) Open image in new window, [ v ( x N ) ] = v ( x N ) Open image in new window and v ( x N ) = v ( x N ) Open image in new window. In case that x k E h Open image in new window are arguments of v ( x k ) Open image in new window or v ( x k + ) Open image in new window, we usually omit these arguments x k Open image in new window, x k + Open image in new window and write simply v Open image in new window and v + Open image in new window, respectively.

3.1 DG semi-discrete formulation

Now, we recall the space semi-discrete DG scheme presented in [11]. First, we multiply (1) by a test function v h S h p Open image in new window, integrate over an element I k T h Open image in new window and use integration by parts in the dispersion term a h Open image in new window and convection term b h ε Open image in new window of (1) subsequently. Further, we sum over all I k T h Open image in new window and add some artificial terms vanishing for the exact solution such as penalty J h σ Open image in new window and stabilization terms, which replace the inter-element discontinuities and guarantee the stability of the resulting numerical scheme, respectively. Consequently, we employ the concept of an upwind numerical flux (see [16]) for the discretization of the convection term and end up with the following DG formulation for the semi-discrete solution u h ( t ) Open image in new window, introduced in [13] as a system of ordinary differential equations, i.e.,
d d t { ( u h ( t ) , v h ) + μ a h ( u h ( t ) , v h ) + μ J h σ ( u h ( t ) , v h ) } + b h ε ( u h ( t ) , v h ) = 0 v h S h , t ( 0 , T ) , Open image in new window
(18)
where forms a h ( , ) Open image in new window and b h ε ( , ) Open image in new window stand for the semi-discrete variants of forms (6) and (7), i.e.,
a h ( u ( t ) , v ) = k J I k u ( t ) x v d x x E h u ( t ) x [ v ] + x E h v [ u ( t ) ] , Open image in new window
(19)
b h ε ( u ( t ) , v ) = k J I k ( u + ε 2 u 2 ) v d x + x E h H ( u ( t ) , u + ( t ) ) [ v ] . Open image in new window
(20)
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 H ( , ) Open image in new window through node x E h Open image in new window in the positive direction (i.e., outer normal is equal to one):
H ( u ( x ) , u ( x + ) ) = { f ( u ( x ) ) = u ( x ) + ε 2 u 2 ( x ) , if  A > 0 , f ( u ( x + ) ) = u ( x + ) + ε 2 u 2 ( x + ) , if  A 0 , Open image in new window
(21)

where A = f ( u ( x ) + u ( x + ) 2 ) Open image in new window and the choice of f ( u ( x 0 ) ) Open image in new window and f ( u ( x N + ) ) Open image in new window for boundary points has to satisfy the prescribed Dirichlet boundary conditions; for more details, see [14].

In what follows, we shall assume that the numerical flux H : R 2 R Open image in new window has the following properties.

Assumptions (H)

(H1) H ( u , v ) Open image in new window is Lipschitz-continuous with respect to u, v:
| H ( u , v ) H ( u , v ) | C H ( | u u | + | v v | ) , u , v , u , v R . Open image in new window
(22)
(H2) H ( u , v ) Open image in new window is consistent:
H ( u , u ) = f ( u ) , u R . Open image in new window
(23)
(H3) H ( u , v ) Open image in new window is conservative:
H ( u , v ) = H ( v , u ) in the negative direction , u , v R . Open image in new window
(24)

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 R 2 Open image in new window.

A particular attention should be also paid to the treatment of the dispersion terms, which include an artificially added stabilization in the form x E h v [ u ( t ) ] Open image in new window, 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.

In the end, the semi-discrete DG scheme is completed with the weighted penalty
J h σ ( u ( t ) , v ) = x E h I σ [ u ( t ) ] [ v ] + σ ( x 0 ) ( u ( x 0 + , t ) u D a ( t ) ) v ( x 0 + ) + σ ( x N ) ( u ( x N , t ) u D b ( t ) ) v ( x N ) Open image in new window
(25)

which replaces the inter-element discontinuities and guarantees the fulfillment of the prescribed boundary conditions.

The penalty parameter function σ : E h R Open image in new window for a nonsymmetric variant is defined in spirit of [17] as
σ ( x ) = { p 1 2 / h 1 , x = a , min ( p k 2 / h k , p k + 1 2 / h k + 1 ) , x E h I { x } = I k I k + 1 , p N 2 / h N , x = b . Open image in new window
(26)
In order to simplify the notation, we introduce the form
A h μ ( u ( t ) , v ) : = ( u ( t ) , v ) + μ a h ( u ( t ) , v ) + μ J h σ ( u ( t ) , v ) , u ( t ) , v S h p , t ( 0 , T ) , Open image in new window
(27)

which is bilinear due to (19) and (25). Consequently, we can here define the semi-discrete solution u h Open image in new window of problem (8).

Definition 2 We say that u h Open image in new window is a semidiscrete solution of problem (8) if u h C 1 ( 0 , T ; S h p ) Open image in new window and the following conditions are satisfied:
(a) d d t A h μ ( u h ( t ) , v ) + b h ε ( u h ( t ) , v h ) = 0 v h S h p , t ( 0 , T ) , (b) ( u h ( 0 ) , v h ) = ( u 0 , v h ) v h S h p . Open image in new window
(28)

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 [13], we have proposed a semi-implicit time discretization based on the backward Euler scheme with the linearized convection form b h ε Open image in new window 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.

We now partition [ 0 , T ] Open image in new window as 0 = t 0 < t 1 < t 2 < < t N = T Open image in new window, denoting each time step by τ l t l + 1 t l Open image in new window and letting u h l Open image in new window stand for the approximate solution of u h ( t l ) Open image in new window, t l [ 0 , T ] Open image in new window, l = 0 , , M Open image in new window. The linearization of the physical flux f is treated in spirit of [13] as
f ( u h l + 1 ) ( 1 + ε u h l ) u h l + 1 ε 2 ( u h l ) 2 , l = 0 , , M , Open image in new window
(29)
which implies the splitting of a convection form in the following way:
b h ε ( u h l + 1 , v h ) b h L ε ( u h l , u h l + 1 , v h ) b h N ε ( u h l , v h ) Open image in new window
(30)
with
b h L ε ( u h l , u h l + 1 , v h ) = k J I k ( 1 + ε u h l ) u h l + 1 v h d x + x E h H L ( ( u h l ) , ( u h l ) + , ( u h l + 1 ) , ( u h l + 1 ) + ) [ v h ] , Open image in new window
(31)
b h N ε ( u h l , v h ) = k J I k ε 2 ( u h l ) 2 v h d x + x E h H N ( ( u h l ) , ( u h l ) + ) [ v h ] , Open image in new window
(32)

where H L ( , , , ) Open image in new window and H N ( , ) Open image in new window represent the corresponding linearized and nonlinear parts of the original numerical flux H ( , ) Open image in new window given by (21); for more details, see [13]. One can easily observe that the form b h L ε ( , , ) Open image in new window is linear in its second and third argument and the form b h N ε ( , ) Open image in new window 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.

Definition 3 Let 0 = t 0 < t 1 < < t r = T Open image in new window be a partition of the time interval [ 0 , T ] Open image in new window and τ l t l + 1 t l Open image in new window, l = 0 , 1 , , M Open image in new window. We define the approximate solution of problem (8) as functions u h l S h p Open image in new window, t [ 0 , T ] Open image in new window, l = 0 , , M Open image in new window, satisfying the following conditions:
(a) A h μ ( u h l + 1 , v h ) + τ l b h L ε ( u h l , u h l + 1 , v h ) = A h μ ( u h l , v h ) + τ l b h N ε ( u h l , v h ) v h S h p , (b) u h 0  is  S h p  approximation of  u 0 . Open image in new window
(33)

Discrete problem (33) is equivalent to a system of linear algebraic equations at each time instant t l [ 0 , T ] Open image in new window. In what follows, we shall be concerned with the analysis of method (33).

Lemma 1 Discrete problem (33) has a unique solution.

Proof We rewrite problem (33) in the following way. For u h l S h p Open image in new window, τ l Open image in new window and t l + 1 [ 0 , T ] Open image in new window, we find u h l + 1 Open image in new window such that
A h l ( u h l + 1 , v h ) = f h l ( v h ) v h S h p , Open image in new window
(34)
where
A h l ( u h l + 1 , v h ) = A h μ ( u h l + 1 , v h ) + τ l b h L ε ( u h l , u h l + 1 , v h ) , u h l , v h S h p Open image in new window
(35)
f h l ( v h ) = A h μ ( u h l , v h ) + τ l b h N ε ( u h l , v h ) , v h S h p . Open image in new window
(36)
Using the definitions (27) and (31), one can see that A h l Open image in new window is a bilinear form on the finite dimensional space and f h l Open image in new window is a linear functional. Moreover, the form A h l Open image in new window is coercive, i.e.,
A h l ( v h , v h ) v h 2 2 v h S h p . Open image in new window
(37)

Hence, equation (34) has a unique solution u h l + 1 S h p Open image in new window. □

4 A priori error analysis

For error analysis and in experiments, we consider p k = p Open image in new window for all k J Open image in new window. Thus we denote S h p = S h p Open image in new window. Now we would like to analyze the error estimates of the approximate solution u h l Open image in new window, l = 0 , 1 , Open image in new window , obtained by method (33). For simplicity, we consider a uniform partition t l = l τ Open image in new window, l = 0 , 1 , , M Open image in new window, of the time interval [ 0 , T ] Open image in new window with time step τ = T / M Open image in new window, where M > 1 Open image in new window is an integer.

Let Π h u l Open image in new window be the standard S h p Open image in new window-interpolation of u l = u ( t l ) Open image in new window, ( l = 0 , , M Open image in new window) satisfying (cf. [18]) for all v H p + 1 ( I k ) Open image in new window, I k T h Open image in new window,
Π h v v L 2 ( I k ) c ˜ h k p + 1 | v | H p + 1 ( I k ) , Open image in new window
(38)
| Π h v v | H 1 ( I k ) c ˜ h k p | v | H p + 1 ( I k ) , Open image in new window
(39)
for a generic constant c ˜ > 0 Open image in new window independent of h and v. We set
ξ h l = u h l Π h u l S h p , η h l = Π h u l u l H p + 1 ( Ω , T h ) . Open image in new window
(40)
Then the error e h l = u h l u l Open image in new window can be expressed as
e h l = ξ h l + η h l , l = 0 , 1 , , M . Open image in new window
(41)
Setting ξ h l + 1 Open image in new window in (33), we get
A h μ ( u h l + 1 u h l , ξ h l + 1 ) + τ ( b h L ε ( u h l , u h l + 1 , ξ h l + 1 ) b h N ε ( u h l , ξ h l + 1 ) ) = 0 . Open image in new window
(42)
On the other hand, from (18) it follows
A h μ ( u t ( t l + 1 ) , ξ h l + 1 ) + b h ε ( u ( t l + 1 ) , ξ h l + 1 ) = 0 . Open image in new window
(43)
Multiplying (43) by τ, subtracting from (42) and using again the linearity of the form A h μ Open image in new window, we get
A h μ ( u h l + 1 u h l τ u t ( t l + 1 ) , ξ h l + 1 ) + τ ( b h L ε ( u h l , u h l + 1 , ξ h l + 1 ) b h N ε ( u h l , ξ h l + 1 ) b h ε ( u ( t l + 1 ) , ξ h l + 1 ) ) = 0 . Open image in new window
(44)
Since u h l + 1 u h l = ξ h l + 1 + η h l + 1 + u l + 1 u l ξ h l η h l Open image in new window, we can rewrite equation (44) in the following way:
A h μ ( ξ h l + 1 ξ h l , ξ h l + 1 ) = A h μ ( u l + 1 u l τ u t ( t l + 1 ) , ξ h l + 1 ) A h μ ( η h l + 1 η h l , ξ h l + 1 ) + τ ( b h L ε ( u h l , u h l + 1 , ξ h l + 1 ) b h N ε ( u h l , ξ h l + 1 ) b h ε ( u ( t l + 1 ) , ξ h l + 1 ) ) . Open image in new window
(45)
For the term on the right-hand side of equation (45), we use decomposition ( m n ) m = 1 2 ( m 2 n 2 + ( m n ) 2 ) Open image in new window and linearity of the form A h μ Open image in new window. Together with (30), we finally get
1 2 { A h μ ( ξ h l + 1 , ξ h l + 1 ) A h μ ( ξ h l , ξ h l ) + A h μ ( ξ h l + 1 ξ h l , ξ h l + 1 ξ h l ) } = A h μ ( u l + 1 u l τ u t ( t l + 1 ) , ξ h l + 1 ) A h μ ( η h l + 1 η h l , ξ h l + 1 ) + τ { b h L ε ( u h l , u h l + 1 , ξ h l + 1 ) b h L ε ( u h l , u ( t l + 1 ) , ξ h l + 1 ) + b h L ε ( u h l , u ( t l + 1 ) , ξ h l + 1 ) b h L ε ( u ( t l + 1 ) , u ( t l + 1 ) , ξ h l + 1 ) + b h N ε ( u ( t l + 1 ) , ξ h l + 1 ) b h N ε ( u h l , ξ h l + 1 ) } . Open image in new window
(46)

For next estimates, we use the following lemmas.

Lemma 2 Under assumptions (R) for t l , t l + 1 [ 0 , T ] Open image in new window, the following hold:
| ( u l + 1 u l τ u t ( t l + 1 ) , ξ h l + 1 ) | c τ 2 ξ h l + 1 2 , Open image in new window
(47)
| ( η h l + 1 η h l , ξ h l + 1 ) | c τ h 2 ( p + 1 ) ξ h l + 1 2 , Open image in new window
(48)
| a h ( u l + 1 u l τ u t ( t l + 1 ) , ξ h l + 1 ) | c τ 2 ( | ξ h l + 1 | 1 , 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) 1 / 2 ) , Open image in new window
(49)
| a h ( η h l + 1 η h l , ξ h l + 1 ) | c τ h 2 ( p + 1 ) ( | ξ h l + 1 | 1 , 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) 1 / 2 ) , Open image in new window
(50)
| J h σ ( u l + 1 u l τ u t ( t l + 1 ) , ξ h l + 1 ) | c τ 2 J h σ ( ξ h l + 1 , ξ h l + 1 ) 1 / 2 , Open image in new window
(51)
| J h σ ( η h l + 1 η h l , ξ h l + 1 ) | c τ h 2 ( p + 1 ) J h σ ( ξ h l + 1 , ξ h l + 1 ) 1 / 2 , Open image in new window
(52)

where c is a generic constant independent of h and τ.

Proof The proof of these standard estimates can be found, for instance, in [15]. □

Lemma 3 Under assumptions (R), (H) and for t l , t l + 1 [ 0 , T ] Open image in new window, the following hold:
| b h L ε ( u h l , u h l + 1 u ( t l + 1 ) , ξ h l + 1 ) | c ( | ξ h l + 1 | 1 , 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) 1 / 2 ) ( h p + 1 + ξ h l + 1 2 ) , Open image in new window
(53)
| b h L ε ( u h l , u ( t l + 1 ) , ξ h l + 1 ) b h L ε ( u ( t l + 1 ) , u ( t l + 1 ) , ξ h l + 1 ) | c ( | ξ h l + 1 | 1 , 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) 1 / 2 ) ( h p + 1 + ξ h l 2 + τ ) , Open image in new window
(54)
| b h N ε ( u ( t l + 1 ) , ξ h l + 1 ) b h N ε ( u h l , ξ h l + 1 ) | c ( | ξ h l + 1 | 1 , 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) 1 / 2 ) ( h p + 1 + ξ h l 2 + τ ) , Open image in new window
(55)

where c is a generic constant independent of h and τ.

Proof Again, one can find the proof of these estimates in [15]. □

Since A h μ ( ξ h l + 1 ξ h l , ξ h l + 1 ξ h l ) Open image in new window is always nonnegative, applying previous lemmas gives us
1 2 A h μ ( ξ h l + 1 , ξ h l + 1 ) 1 2 A h μ ( ξ h l , ξ h l ) c { τ ( τ + h p + 1 ) ξ h l + 1 2 + μ τ ( τ + h p ) ( | ξ h l + 1 | 1 , 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) 1 / 2 ) + μ τ ( τ + h p ) J h σ ( ξ h l + 1 , ξ h l + 1 ) 1 / 2 + τ ( | ξ h l + 1 | 1 , 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) 1 / 2 ) ( ξ h l + 1 2 + h p + 1 + ξ h l 2 + τ ) } . Open image in new window
(56)
Multiplying by 2, applying the Young inequality and using the definition of the form A h μ Open image in new window, we obtain
ξ h l + 1 2 2 + μ ( | ξ h l + 1 | 1 , 2 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) ) ξ h l 2 2 + μ ( | ξ h l | 1 , 2 2 + J h σ ( ξ h l , ξ h l ) ) + 2 c τ { τ 2 2 ν 1 + ν 1 2 ξ h l + 1 2 2 + h 2 ( p + 1 ) 2 ν 2 + ν 2 2 ξ h l + 1 2 2 + μ ( τ 2 ν 3 + h 2 ( p ) ν 4 + ν 3 + ν 4 2 ( | ξ h l + 1 | 1 , 2 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) ) ) + μ ( τ 2 2 ν 5 + h 2 ( p ) 2 ν 6 + ν 5 + ν 6 2 J h σ ( ξ h l + 1 , ξ h l + 1 ) ) + ν 7 + ν 8 + ν 9 + ν 10 2 μ ( | ξ h l + 1 | 1 , 2 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) ) + 1 μ ν 7 ξ h l + 1 2 2 + h 2 ( p + 1 ) μ ν 8 + 1 μ ν 9 ξ h l 2 2 + τ 2 μ ν 10 } . Open image in new window
(57)
If we take into account that J h σ ( ξ h l + 1 , ξ h l + 1 ) | ξ h l + 1 | 1 , 2 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) Open image in new window, the previous inequality can be rewritten as
( 1 c τ ( ν 12 + 2 μ ν 7 ) ) ξ h l + 1 2 2 + ( 1 c τ i = 3 10 ν i ) μ ( | ξ h l + 1 | 1 , 2 2 + J h σ ( ξ h l + 1 , ξ h l + 1 ) ) ( 1 + 2 c τ ν 9 μ ) ξ h l 2 2 + μ ( | ξ h l | 1 , 2 2 + J h σ ( ξ h l , ξ h l ) ) + c τ q ( τ , h , μ ) , Open image in new window
(58)
where we denoted ν 12 = ν 1 + ν 2 Open image in new window and
q ( τ , h , μ ) = ( 1 ν 1 + 2 μ ν 3 + μ ν 5 + 2 μ ν 10 ) τ 2 + ( 2 ν 4 + 1 2 ν 6 ) μ h 2 p + ( 1 ν 2 + 2 ν 8 μ ) h 2 ( p + 1 ) . Open image in new window
(59)
Let us now introduce the so-called energy norm
v h = | v h | 1 , 2 2 + J h σ ( v h , v h ) Open image in new window
(60)
and the norm
v h μ = v h 2 2 + μ v h 2 . Open image in new window
(61)
Denoting C L = c max { ν 12 + 2 μ ν 7 , i = 3 10 ν i } Open image in new window and C R = 2 c τ μ ν 9 Open image in new window from (58), it follows
( 1 τ C L ) ξ h l + 1 μ 2 ( 1 + C R ) ξ h l μ 2 + c τ q ( τ , h , μ ) . Open image in new window
(62)

In order to finish our estimates, we require a fulfillment of the following technical assumption.

Assumption (T)

(T1) There exists θ ( 0 , 1 ) Open image in new window such that 0 < τ < θ / C L Open image in new window.

If assumption (T) is fulfilled, then τ < 1 C L μ μ ν 12 + 2 / ν 7 ν 7 2 μ Open image in new window. Thus we can also reformulate assumption (T) so that τ = O ( μ ) Open image in new window.

Thus, let us assume that assumption (T) holds, then
ξ h l + 1 μ 2 B ξ h l μ 2 + c τ 1 τ C L q ( τ , h , μ ) Open image in new window
(63)
with B = 1 + τ C R 1 τ C L = 1 + τ C L + C R 1 τ C L exp ( τ C L + C R 1 τ C L ) Open image in new window. Consequently,
ξ h l μ 2 B l ξ h 0 μ 2 + B l 1 B 1 c τ 1 τ C L q ( τ , h , μ ) Open image in new window
(64)
and since B 1 = τ C L + C R 1 τ C L Open image in new window, we have c τ ( B 1 ) ( 1 τ C L ) = c C L + C R Open image in new window, i.e.,
ξ h l μ 2 B l ξ h 0 μ 2 + ( B l 1 ) c C L + C R q ( τ , h , μ ) exp ( l τ C L + C R 1 τ C L ) ( ξ h 0 μ 2 + c C L + C R q ( τ , h , μ ) ) C ˜ exp ( T C L + C R 1 τ C L ) ( μ h 2 p + h 2 ( p + 1 ) + μ ν 7 ν 9 2 ( ν 7 + ν 9 ) q ( τ , h , μ ) ) , Open image in new window
(65)

where we used a straightforward estimate ξ h 0 μ 2 C ˜ ( μ h 2 p + h 2 ( p + 1 ) ) Open image in new window. We can notice that due to the presence of the factor μ in front of the function q ( τ , h , μ ) Open image in new window on the left-hand side of (65), we lost μ in denominators of q ( τ , h , μ ) Open image in new window.

Now we are ready to formulate the main theorem.

Theorem 1 Let assumptions (M), (H), (R) and (T) be satisfied, then there exists a constant C = C ( μ ) Open image in new window such that

where Open image in new window is defined by (60).

Proof Since e h l μ ξ h l μ + η h l μ Open image in new window, the statement of the theorem comes from (65) and the fact that η h l μ c ˜ ( h p + 1 + μ h p ) Open image in new window. Then we set
C ( μ ) = C ˜ exp ( T C L + C R 1 θ ) ν 7 ν 9 ν 7 + ν 9 max { 1 ν j , j { 1 , 2 , , 6 , 8 , 10 } } . Open image in new window
(68)

 □

Remark 3 Theorem 1 implies that the error of our method is O ( h p + τ ) Open image in new window in both energy and L 2 Open image in new window-norm. However, as we will see in the next section, the error estimate in the L 2 Open image in new window-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 [19]. This is useful namely in the cases when convection terms dominate, i.e., μ 0 + Open image in new window. Consequently, in these cases assumption (T) can be weakened to a CFL-like condition τ = O ( h α ) Open image in new window for suitable α > 0 Open image in new window.

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.

We verify numerically the convergence of the method in the L 2 Open image in new window-norm and the energy norm given by (60) with respect to time step τ and mesh size h. The computational errors are evaluated at certain time instants t = l τ Open image in new window during all computations in the corresponding norms, i.e.,
err h , τ 0 , l u h l u ( l τ ) 2 , Open image in new window
(69)
err h , τ 1 , l u h l u ( l τ ) , Open image in new window
(70)
where u ( l τ ) Open image in new window is a prescribed exact solution at time and u h l Open image in new window is the numerical solution at time level obtained by the semi-implicit scheme (33) with constant time step τ on the uniform grid with mesh size h. We suppose that errors behave according to the formula
err h , τ n = err h n + err τ n , n = 0 , 1 , Open image in new window
(71)
where
err h n D n ˜ h a n , err τ n D n ˆ τ b n , n = 0 , 1 . Open image in new window
(72)
The constants D n ˜ Open image in new window, n = 0 , 1 Open image in new window, are independent of τ and D n ˆ Open image in new window, n = 0 , 1 Open image in new window, are independent of h. The values a n Open image in new window, b n Open image in new window, n = 0 , 1 Open image in new window, are the orders of accuracy of the method in the corresponding considered norms. We define the experimental order of convergence (EOC) by
a n = log ( err h 1 , τ n / err h 2 , τ n ) log ( h 1 / h 2 ) and b n = log ( err h , τ 1 n / err h , τ 2 n ) log ( τ 1 / τ 2 ) , n = 0 , 1 . Open image in new window
(73)

5.1 Single solitary case

The RLW equation has the following analytical single solitary wave solution given by
u ( x , t ) = 3 c sech 2 ( B 0 ( x x ¯ v t ) ) with  B 0 = 1 2 ε c μ ( 1 + ε c ) , Open image in new window
(74)

which represents a single solitary wave of amplitude 3c, traveling with the velocity v = 1 + ε c Open image in new window in a positive x-direction and located initially at the point x ¯ Open image in new window. The initial condition is extracted from the exact solution (74) and homogeneous Dirichlet boundary conditions are set.

In order to compare our semi-implicit approach to the schemes given in [6, 20] and [5], we set the parameter values c = 0.1 Open image in new window, x ¯ = 0.0 Open image in new window, ε = μ = 1.0 Open image in new window. The run of the algorithm is carried out up to time T = 20.0 Open image in new window over the problem domain [ 40 , 60 ] Open image in new window. The resulting linear algebraic problems (33) are solved by the restarted GMRES method. Figure 1 captures the development of approximation solutions of a single solitary wave from an initial condition to the terminal time T for a piecewise linear approximation with time step τ = 0.001 Open image in new window and mesh size h = 0.05 Open image in new window. The similar plots are also obtained for another combination of τ and h as we consider below.
Figure 1

The 3D plot of approximation solutions of a single solitary wave (left) and corresponding isolines in space-time 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 τ = 10 3 Open image in new window. Numerical experiments are carried out with the use of piecewise linear ( P 1 Open image in new window) approximations on five consecutive uniformly refined meshes having 125, 250, 500, 1,000 and 2,000 elements.

Tables 1 and 2 show computational errors in the L 2 Open image in new window-norm and the energy norm at four time instances t = 5.0 Open image in new window, t = 10.0 Open image in new window, t = 15.0 Open image in new window and t = 20.0 Open image in new window, the corresponding EOC during all computations. Since the exact solution u ( t ) Open image in new window is sufficiently regular over Ω, it follows from Remark 3 that the theoretical error estimates are of order O ( h p + τ ) Open image in new window. On the other hand, we observe that the numerical experiment of propagation of a single solitary wave indicates a better behavior of EOC in the L 2 Open image in new window-norm, which is expected to be asymptotically O ( h 2 ) Open image in new window for piecewise linear ( p = 1 Open image in new window) approximations. These observations also correspond with the finite element approach from [6], where the same example was studied.
Table 1

Single solitary case: Computational errors in the L 2 Open image in new window -norm and experimental orders of convergence for P 1 Open image in new window approximation on a consequence of meshes at time instances t ( τ = 10 3 Open image in new window )

h

t = 5.0 Open image in new window

t = 10.0 Open image in new window

t = 15.0 Open image in new window

t = 20.0 Open image in new window

err h 0 Open image in new window

EOC

err h 0 Open image in new window

EOC

err h 0 Open image in new window

EOC

err h 0 Open image in new window

EOC

0.80

2.643E-03

-

3.507E-03

-

4.376E-03

-

5.255E-03

-

0.40

6.732E-04

1.973

8.762E-04

2.001

1.099E-03

1.993

1.298E-03

2.017

0.20

1.634E-04

2.043

2.232E-04

1.973

2.605E-04

2.077

3.284E-04

1.983

0.10

4.061E-05

2.009

5.763E-05

1.953

6.561E-05

1.989

8.061E-05

2.026

0.05

1.004E-05

2.016

1.355E-05

2.089

1.638E-05

2.002

2.107E-05

1.936

Table 2

Single solitary case: Computational errors in the energy norm and experimental orders of convergence for P 1 Open image in new window approximation on a consequence of meshes at time instances t ( τ = 10 3 Open image in new window )

h

t = 5.0 Open image in new window

t = 10.0 Open image in new window

t = 15.0 Open image in new window

t = 20.0 Open image in new window

err h 1 Open image in new window

EOC

err h 1 Open image in new window

EOC

err h 1 Open image in new window

EOC

err h 1 Open image in new window

EOC

0.80

7.082E-03

-

7.096E-03

-

7.108E-03

-

7.123E-03

-

0.40

3.540E-03

1.000

3.550E-03

0.999

3.603E-03

0.980

3.673E-03

0.956

0.20

1.778E-03

0.993

1.791E-03

0.987

1.810E-03

0.993

1.891E-03

0.958

0.10

8.866E-04

1.004

8.943E-04

1.002

9.049E-04

1.000

9.099E-04

1.055

0.05

4.441E-04

0.997

4.492E-04

0.993

4.533E-04

0.997

4.573E-04

0.993

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 O ( h p ) Open image in new window. Finally, both estimates in Theorem 1 confirm the well-know attribute of DG schemes from the class of convection-diffusion problems, cf. [14] and [15].

5.1.2 Convergence with respect to τ

Secondly, we verify experimentally the convergence of the method in the L 2 Open image in new window-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.

The computations were carried out with five different time steps τ, see Table 3. The computational error is evaluated at final time t = T Open image in new window in the L 2 Open image in new window-norm and the energy norm, respectively. We observe that both computational errors have EOC of order O ( τ ) Open image in new window in the corresponding norms, which is again in a good agreement with derived theoretical results.
Table 3

Single solitary case: Computational errors in the L 2 Open image in new window -norm and the energy norm for P 1 Open image in new window approximation with respect to time step ( h = 0.05 Open image in new window )

τ

t = 20.0 Open image in new window

t = 20.0 Open image in new window

err τ 0 Open image in new window

EOC

err τ 1 Open image in new window

EOC

0.2000

5.852E-02

-

8.647E-03

-

0.1000

2.977E-02

0.975

4.384E-03

1.008

0.0500

1.445E-02

1.046

2.160E-03

0.994

0.0250

7.228E-03

0.996

1.097E-03

0.977

0.0125

3.601E-03

1.005

5.300E-04

1.049

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
I M ( u ) = Ω u d x , Open image in new window
(75)
momentum
I P ( u ) = Ω ( u 2 + μ ( u ) 2 ) d x , Open image in new window
(76)
and energy
I E ( u ) = Ω ( u 3 + 3 u 2 ) d x , Open image in new window
(77)
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
I M ( u ) = 6 c B 0 , I P ( u ) = 12 c 2 B 0 + 48 B 0 c 2 μ 5 , I E ( u ) = 36 c 2 B 0 + 144 c 3 5 B 0 . Open image in new window
(78)
Moreover, for the purpose of a more accurate comparison with reference results, we introduce the discrete l Open image in new window-norm defined by
err , l max k J ( | u h l ( x k 1 + ) u ( x k 1 + , l τ ) | , | u h l ( x k ) u ( x k , l τ ) | ) Open image in new window
(79)

assessing the accuracy of the method by measuring the difference between the numerical and analytic solutions u h Open image in new window and u, respectively.

Table 4 records the invariant quantities together with errors err Open image in new window, err 0 Open image in new window computed on the finest space-time grid and compares obtained results with several previously presented schemes given in [6, 7, 20] and [5]. All three conservation quantities are kept almost constant thus they illustrate the suitability of the proposed scheme for this problem. The obtained satisfactory results correspond to the reference ones from [6, 7, 20] and [5]. Moreover, we append the recent results from [13] in order to compare the incomplete variant of stabilization in dispersion terms with the nonsymmetric one.
Table 4

Single solitary case: Computed invariant quantities and errors in the l Open image in new window -norm and the L 2 Open image in new window -norm ( h = 0.05 Open image in new window , τ = 10 3 Open image in new window )

Method

Time

err Open image in new window

err 0 Open image in new window

I M ( u h l ) Open image in new window

I P ( u h l ) Open image in new window

I E ( u h l ) Open image in new window

present method

0.0

-

-

3.9799

0.8105

2.5790

5.0

2.325E-05

1.004E-05

3.9799

0.8104

2.5787

10.0

4.518E-05

1.355E-05

3.9799

0.8103

2.5785

15.0

6.901E-05

1.638E-05

3.9799

0.8101

2.5782

20.0

9.322E-05

2.107E-05

3.9799

0.8100

2.5780

ref. meth. [6] (h = 0.125, τ = 0.1)

20.0

6.843E-04

1.757E-03

3.9800

0.8104

2.5792

ref. meth. [7] (h = 0.1, τ = 0.1)

20.0

1.501E-03

1.480E-05

3.96467

0.80462

2.56972

ref. meth. [20] (h = 0.8, τ = 0.1)

20.0

6.660E-05

1.820E-04

3.97992

0.81046

2.57901

ref. meth. [13] (h = 0.125, τ = 0.1)

20.0

3.960E-03

9.092E-03

3.9800

0.8105

2.5791

ref. meth. [5] (h = 0.05, τ = 0.1)

20.0

7.805E-05

2.069E-04

3.97988

0.81046

2.57901

analytical val. (c = 0.1, μ = 1.0)

-

-

-

3.979949

0.810462

2.579007

5.2 Periodic case

The family of periodic solutions of the RLW equation may be analytically written as (cf. [20])
u ( x , t ) = A 0 + A 1 dn 2 ( B 1 ( x x ¯ v t ) , k ) with  v = 1 + ε c Open image in new window
(80)
and the parameters A 0 Open image in new window, A 1 Open image in new window and B 1 Open image in new window given by
A 0 = c ( 1 2 k 2 k 4 k 2 + 1 ) , A 1 = 3 c k 4 k 2 + 1 , Open image in new window
(81)
B 1 = 1 2 ε c μ ( 1 + ε c ) 1 k 4 k 2 + 1 4 , Open image in new window
(82)
where dn ( , k ) Open image in new window is the Jacobi elliptic function and k [ 0 , 1 ) Open image in new window stands for the elliptic modulus; for definitions and other properties, see [21]. The exact solution (80)-(82) represents a one-parameter family of periodic waves of amplitude A 0 + A 1 Open image in new window, traveling with the velocity v in a positive x-direction. The spatial period ω k Open image in new window and time period T k Open image in new window for each wave are defined by
ω k = 2 K ( k ) / B 1 and T k = ω k / v , Open image in new window
(83)

where K ( k ) Open image in new window is a complete elliptic integral of the first kind, see [21]. The limit k 1 Open image in new window 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 c = 0.1 Open image in new window, x ¯ = 0.0 Open image in new window, ε = μ = 1.0 Open image in new window and the parameter k is experimentally set up as k = 0.63048 Open image in new window to have periods T k 20.0 Open image in new window and ω k 22.0 Open image in new window.

The run of the algorithm is carried out up to one time period T k Open image in new window over the problem domain [ 2 ω k , 2 ω k ] Open image in new window. The initial and nonhomogeneous Dirichlet conditions are extracted from the exact solution (80) and the same linear algebraic solver is used as in the previous case. Figure 2 depicts the propagation of approximation solutions of periodic waves from the initial condition to the final time T k Open image in new window for a piecewise linear approximation on the finest considered space-time mesh with time step τ = 0.001 Open image in new window and mesh size h = 0.05 Open image in new window. Other coarse grids also produce similar plots.
Figure 2

The 3D plot of approximation solutions of periodic waves (left) and corresponding isolines in space-time 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 h-convergence in the periodic case is investigated on a sequence of five successive refined grids partitioning the considered problem domain [ 44 , 44 ] Open image in new window. 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.

The obtained results recorded in Tables 5 and 6 illustrate the same behavior of computational errors in the L 2 Open image in new window-norm and the energy norm with respect to the spatial discretization as in the case of a single solitary wave propagation. The computed EOCs at all four monitoring time instances keep asymptotically the same orders, i.e., err h 0 = O ( h 2 ) Open image in new window and err h 1 = O ( h ) Open image in new window, for piecewise linear approximations and confirm the spatially suboptimal a priori error estimates (66) with respect to the L 2 Open image in new window-norm and spatially optimal estimates (67) in the energy norm, respectively. The influence of discretization errors on computations with a long time domain can be better eliminated by using the Crank-Nicolson numerical scheme instead of the backward Euler method.
Table 5

Periodic case: Computational errors in the L 2 Open image in new window -norm and experimental orders of convergence for P 1 Open image in new window approximation on a consequence of meshes at time instances t ( τ = 10 3 Open image in new window )

h

t = 5.0 Open image in new window

t = 10.0 Open image in new window

t = 15.0 Open image in new window

t = 20.0 Open image in new window

err h 0 Open image in new window

EOC

err h 0 Open image in new window

EOC

err h 0 Open image in new window

EOC

err h 0 Open image in new window

EOC

0.80

2.790E-02

-

3.589E-02

-

4.531E-02

-

6.290E-02

-

0.40

7.016E-03

1.992

9.085E-03

1.982

1.156E-02

1.971

1.612E-03

1.964

0.20

1.860E-03

1.915

2.308E-03

1.977

3.005E-03

1.943

4.281E-03

1.912

0.10

4.952E-04

1.909

6.179E-04

1.901

8.084E-04

1.894

1.204E-04

1.830

0.05

1.357E-04

1.868

1.724E-04

1.841

2.314E-04

1.804

3.534E-04

1.768

Table 6

Periodic case: Computational errors in the energy norm and experimental orders of convergence for P 1 Open image in new window approximation on a consequence of meshes at time instances t ( τ = 10 3 Open image in new window )

h

t = 5.0 Open image in new window

t = 10.0 Open image in new window

t = 15.0 Open image in new window

t = 20.0 Open image in new window

err h 1 Open image in new window

EOC

err h 1 Open image in new window

EOC

err h 1 Open image in new window

EOC

err h 1 Open image in new window

EOC

0.80

9.029E-03

-

9.004E-03

-

9.115E-03

-

9.251E-03

-

0.40

4.615E-03

0.968

4.658E-03

0.951

4.767E-03

0.935

4.862E-03

0.928

0.20

2.370E-03

0.961

2.488E-03

0.905

2.597E-03

0.876

2.770E-03

0.812

0.10

1.275E-03

0.894

1.363E-03

0.868

1.502E-03

0.790

1.601E-03

0.790

0.05

6.906E-04

0.885

7.533E-04

0.856

8.569E-04

0.809

9.154E-04

0.807

5.2.2 Convergence with respect to τ

The τ-convergence 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 T k Open image in new window. The theoretical results are in accordance with the observations listed in Table 7, i.e., err τ 0 = O ( h ) Open image in new window and err τ 1 = O ( h ) Open image in new window.
Table 7

Periodic case: Computational errors in the L 2 Open image in new window -norm and the energy norm for P 1 Open image in new window approximation with respect to time step ( h = 0.05 Open image in new window )

τ

t = 20.0 Open image in new window

t = 20.0 Open image in new window

err τ 0 Open image in new window

EOC

err τ 1 Open image in new window

EOC

0.2000

7.690E-02

-

2.287E-02

-

0.1000

4.061E-02

0.921

1.221E-02

0.905

0.0500

2.078E-02

0.967

6.393E-03

0.934

0.0250

1.044E-02

0.993

3.453E-03

0.889

0.0125

5.514E-03

0.921

1.831E-03

0.915

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 [ 2 ω k , 2 ω k ] Open image in new window, all these three invariants of motion remain conserved and equal to their original values that are well-determined analytically at t = 0 Open image in new window.

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 l Open image in new window-norm and the L 2 Open image in new window-norm and keeping the invariant quantities during the whole computation performed on the finest space-time grid. All three invariants of motion are not different from their analytical values, according to which this method can be considered suitable also for nonperiodic cases.
Table 8

Periodic case: Computed invariant quantities and errors in the l Open image in new window -norm and the L 2 Open image in new window -norm ( h = 0.05 Open image in new window , τ = 10 3 Open image in new window )

Method

Time

err Open image in new window

err 0 Open image in new window

I M ( u h l ) Open image in new window

I P ( u h l ) Open image in new window

I E ( u h l ) Open image in new window

present method

0.0

-

-

16.5051

3.3180

10.6008

5.0

2.855E-05

1.357E-04

16.5056

3.3180

10.6005

10.0

3.063E-05

1.724E-04

16.5057

3.3179

10.6003

15.0

4.604E-05

2.314E-04

16.5057

3.3178

10.6000

20.0

7.854E-05

3.534E-04

16.5060

3.3178

10.6001

analytical val. (k = 0.63048)

-

-

-

16.50560

3.318064

10.60086

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 semi-implicit time discretization with suitable linearization of convective terms. Under some additional assumptions, we have derived a priori error estimates, namely O ( h p + τ ) Open image in new window in the L 2 Open image in new window-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 ( L 2 ) Open image in new window-order of convergence, which is expected to be asymptotically O ( h 2 + τ ) Open image in new window 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.

Notes

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.

Supplementary material

13661_2012_374_MOESM1_ESM.gif (37 kb)
Authors’ original file for figure 1
13661_2012_374_MOESM2_ESM.gif (58 kb)
Authors’ original file for figure 2
13661_2012_374_MOESM3_ESM.eps (677 kb)
Authors’ original file for figure 3
13661_2012_374_MOESM4_ESM.eps (271 kb)
Authors’ original file for figure 4

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Copyright information

© Hozman and Lamač; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Authors and Affiliations

  1. 1.Faculty of Science, Humanities and EducationTechnical University of LiberecLiberecCzech Republic
  2. 2.Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic

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